---
title: "Handbook of Magma Functions, Part X: Finitely Presented Groups"
version: "2.30-1"
canonical: "https://docs.magma-maths.org/FinitelyPresentedGroups/index-finitely-presented-groups.html"
---
# Finitely Presented Groups
- [Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md): Abelian group creation, generic abelian groups, and elements.
- [Free Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/index-free-groups.md): Free group creation, subgroups, and automorphisms.
- [Introduction to FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/index-introduction-to-fpgroups.md): FP-group construction, properties, and subgroups.
- [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md): FP-group construction, simplification, and standard constructions.
- [Polycyclic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/PolycyclicGroups/index-polycyclic-groups.md): Polycyclic group construction, subgroups, quotients, and conversion.
- [Braid Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/BraidGroups/index-braid-groups.md): Braid group construction and element creation.
- [Groups Defined by Rewrite Systems](https://docs.magma-maths.org/FinitelyPresentedGroups/GroupsDefinedByRewriteSystems/index-groups-defined-by-rewrite-systems.md): Rewriting system creation, properties, and arithmetic for groups.
- [Automatic and Hyperbolic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AutomaticAndHyperbolicGroups/index-automatic-and-hyperbolic-groups.md): Automatic groups and hyperbolic groups.
- [Groups of Straight-line Programs](https://docs.magma-maths.org/FinitelyPresentedGroups/GroupsOfStraightLinePrograms/index-groups-of-straight-line-programs.md): Straight-line program group creation and operations.
- [Finitely Presented Semigroups](https://docs.magma-maths.org/FinitelyPresentedGroups/FinitelyPresentedSemigroups/index-finitely-presented-semigroups.md): Semigroup presentation creation, operations, and properties.
- [Monoids Given by Rewrite Systems](https://docs.magma-maths.org/FinitelyPresentedGroups/MonoidsGivenByRewriteSystems/index-monoids-given-by-rewrite-systems.md): Rewriting system creation, operations, and homomorphisms for monoids.
## Abelian Groups
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/introduction.md)
- [Construction of a Finitely Presented Abelian Group and its Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md)
- [The Free Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#the-free-abelian-group)
- [`FreeAbelianGroup(n): RngIntElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-freeabeliangroup-rngintelt)
- [Example: Free Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#example-ex-c01eb3)
- [Relations](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#relations)
- [`w₁ = w₂: GrpAbElt, GrpAbElt → Rel`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#operation-op-grpabelt-grpabelt)
- [`r[1]: GrpAbRel, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#indexed-indexed-r-grpabrel-rngintelt-grpabelt)
- [`LHS(r): Rel → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-lhs-rel)
- [`r[2]: GrpAbRel, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#indexed-indexed-r-grpabrel-rngintelt-grpabelt-2)
- [`RHS(r): Rel → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-rhs-rel)
- [`Parent(r): RelElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-parent-relelt)
- [Example: Relations in a Free Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#example-ex-0f1824)
- [Specification of a Presentation](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#specification-of-a-presentation)
- [`AbelianGroup< X | R >: List(Var), List(GrpAbRel) → GrpAb, Hom(GrpAb)`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#constructor-constructor-abeliangroup-list-var-list-grpabrel-grpab-hom-grpab)
- [Example: Abelian Group from a Presentation](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#example-ex-bd5820)
- [`AbelianGroup([n₁,...,nᵣ]): [ RngIntElt ] → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-abeliangroup-rngintelt)
- [Example: Abelian Group from Cyclic Factors](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#example-ex-ea4c01)
- [Accessing the Defining Generators and Relations](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#accessing-the-defining-generators-and-relations)
- [`A . i: GrpAb, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#operation-operation-grpab-rngintelt-grpabelt)
- [`Generators(A): GrpAb → { GrpAbElt}`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-generators-grpab)
- [`NumberOfGenerators(A): GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-numberofgenerators-grpab)
- [`Ngens(A): GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-ngens-grpab)
- [`Parent(u): GrpAbElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-parent-grpabelt)
- [`Relations(A): GrpAb → [ Rel ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-relations-grpab)
- [`RelationMatrix(A): GrpAb → Mtrx`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/creation.md#function-relationmatrix-grpab)
- [Construction of a Generic Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md)
- [Specification of a Generic Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#specification-of-a-generic-abelian-group)
- [`GenericAbelianGroup(U: parameters): . → GrpAbGen`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#function-genericabeliangroup)
- [Example: Generic Abelian Group of Units](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#example-ex-5df08f)
- [Accessing Generators](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#accessing-generators)
- [`Universe(A): GrpAbGen`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#procedure-universe-grpabgen)
- [`A . i: GrpAbGen, RngIntElt → GrpAbGenElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#operation-operation-grpabgen-rngintelt-grpabgenelt)
- [`Generators(A): GrpAbGen → [ GrpAbGenElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#function-generators-grpabgen)
- [`UserGenerators(A): GrpAbGen → [ GrpAbGenElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#function-usergenerators-grpabgen)
- [`NumberOfGenerators(A): GrpAbGen → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#function-numberofgenerators-grpabgen)
- [`Ngens(A): GrpAbGen → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#function-ngens-grpabgen)
- [Computing Abelian Group Structure](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#computing-abelian-group-structure)
- [`AbelianGroup(A: parameters): GrpAbGen → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#function-abeliangroup-grpabgen)
- [Example: Structure of a Unit Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#example-ex-0d76ee)
- [Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md)
- [Construction of Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#construction-of-elements)
- [`A ! [a₁, ... ,aₙ]: GrpAb, [RngIntElt] → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-grpab-rngintelt)
- [`A ! [a₁, ... ,aₙ]: GrpAbGen, [RngIntElt] → GrpAbGenElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-grpabgen-rngintelt)
- [`A ! e: GrpAbGen, Elt → GrpAbGenElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-grpabgen-elt)
- [`A ! g: GrpAbGen, GrpAbGenElt → GrpAbGenElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-grpabgen-grpabgenelt)
- [`A ! n: GrpAb, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-grpab-rngintelt-2)
- [`Random(A): GrpAbGen → GrpAbGenElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-random-grpabgen)
- [`Identity(A): GrpAb → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-identity-grpab)
- [`Id(A): GrpAb → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-id-grpab)
- [`A ! 0: GrpAb, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-grpab-rngintelt-3)
- [Representation of an Element](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#representation-of-an-element)
- [`Representation(g): GrpAbGenElt → [RngIntElt]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-representation-grpabgenelt)
- [`ElementToSequence(g): GrpAbGenElt → [RngIntElt]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-elementtosequence-grpabgenelt)
- [`Eltseq(g): GrpAbGenElt → [RngIntElt]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-eltseq-grpabgenelt)
- [`UserRepresentation(g): GrpAbGenElt → [RngIntElt]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-userrepresentation-grpabgenelt)
- [`Representation(S, g): SeqEnum, GrpAbGenElt → [RngIntElt], RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#function-representation-seqenum-grpabgenelt)
- [Example: Creating and Representing Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#example-ex-f86c65)
- [Arithmetic with Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#arithmetic-with-elements)
- [`u + v: GrpAbElt, GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-plus-grpabelt-grpabelt)
- [`- u: GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-operation-grpabelt-grpabelt)
- [`u - v: GrpAbElt, GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-minus-grpabelt-grpabelt)
- [`m * u: RngIntElt, GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-times-rngintelt-grpabelt)
- [`u * m: GrpAb, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/elements.md#operation-op-times-grpab-rngintelt)
- [Construction of Subgroups and Quotient Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md)
- [Construction of Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#construction-of-subgroups)
- [`sub: GrpAb, List → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#constructor-constructor-sub-grpab-list-grpab-map)
- [Example: Creating a Subgroup of an Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#example-ex-f256c6)
- [`sub: GrpAbGen, List → GrpAbGen`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#constructor-constructor-sub-grpabgen-list-grpabgen)
- [Example: Subgroup of a Quadratic Forms Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#example-ex-8e5822)
- [Construction of Quotient Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#construction-of-quotient-groups)
- [`quo: GrpAb, List → GrpAb, Hom(GrpAb)`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#constructor-constructor-quo-grpab-list-grpab-hom-grpab)
- [`A / B: GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-quotient.md#operation-op-div-grpab-grpab)
- [Standard Constructions and Conversions](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md)
- [`AbelianGroup(GrpAb, Q): Cat, [ RngIntElt ] → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-abeliangroup-cat-rngintelt)
- [`AbelianGroup(Q): [ RngIntElt ] → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-abeliangroup-rngintelt-2)
- [`AbelianGroup(G): Grp → GrpAb, Hom`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-abeliangroup-grp)
- [`AbelianQuotient(G): Grp → GrpAb, Hom`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-abelianquotient-grp)
- [`DirectSum(A, B): GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-directsum-grpab-grpab)
- [`PCGroup(A): GrpAb → GrpPC, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-pcgroup-grpab)
- [`PermutationGroup(A): GrpAb → GrpPerm, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-permutationgroup-grpab)
- [`FPGroup(A): GrpAb → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-fpgroup-grpab)
- [`CommutatorSubgroup(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-commutatorsubgroup-grpab)
- [`DerivedSubgroup(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-derivedsubgroup-grpab)
- [`DerivedGroup(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-derivedgroup-grpab)
- [`CommutatorSubgroup(H, K): GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-commutatorsubgroup-grpab-grpab)
- [`CommutatorSubgroup(G, H, K): GrpAb, GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-commutatorsubgroup-grpab-grpab-grpab)
- [`Centralizer(G, a): GrpAb, GrpAbElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-centralizer-grpab-grpabelt)
- [`Centraliser(G, a): GrpAb, GrpAbElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-centraliser-grpab-grpabelt)
- [`Core(G, H): GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-core-grpab-grpab)
- [`Centre(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-centre-grpab)
- [`Center(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-center-grpab)
- [`FittingGroup(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-fittinggroup-grpab)
- [`FittingSubgroup(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-fittingsubgroup-grpab)
- [`Hypercentre(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-hypercentre-grpab)
- [`Hypercenter(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/standard-construction.md#function-hypercenter-grpab)
- [Operations on Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md)
- [Order of an Element](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#order-of-an-element)
- [`Order(x): GrpAbElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-order-grpabelt)
- [Example: Orders of Abelian Group Elements](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#example-ex-5056af)
- [`Order(g: parameters): GrpAbGenElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-order-grpabgenelt)
- [`Order(g, l, u: parameters): GrpAbGenElt, RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-order-grpabgenelt-rngintelt-rngintelt)
- [`Order(g, l, u, n, m: parameters): GrpAbGenElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-order-grpabgenelt-rngintelt-rngintelt-rngintelt-rngintelt)
- [Discrete Logarithm](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#discrete-logarithm)
- [`Log(g, d: parameters): GrpAbGenElt, GrpAbGenElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-log-grpabgenelt-grpabgenelt)
- [Example: Discrete Logarithm for Quadratic Forms](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#example-ex-cb963b)
- [Equality and Comparison](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#equality-and-comparison)
- [`u eq v: GrpAbElt, GrpAbElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#operation-op-eq-grpabelt-grpabelt)
- [`u ne v: GrpAbElt, GrpAbElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#operation-op-ne-grpabelt-grpabelt)
- [`IsIdentity(u): GrpAbElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-isidentity-grpabelt)
- [`IsId(u): GrpAbElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/operation-element.md#function-isid-grpabelt)
- [Invariants of an Abelian Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md)
- [`ElementaryAbelianQuotient(G, p): GrpAb, RngIntElt → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-elementaryabelianquotient-grpab-rngintelt)
- [`FreeAbelianQuotient(G): GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-freeabelianquotient-grpab)
- [`PrimaryAbelianInvariants(A): GrpAb → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-primaryabelianinvariants-grpab)
- [`AbelianInvariants(A): GrpAb → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-abelianinvariants-grpab)
- [`PrimaryAbelianBasis(A): GrpAb → [ GrpAbElt ], [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-primaryabelianbasis-grpab)
- [`AbelianBasis(A): GrpAb → [ GrpAbElt ], [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-abelianbasis-grpab)
- [`TorsionFreeRank(A): GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-torsionfreerank-grpab)
- [`TorsionInvariants(A): GrpAb → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-torsioninvariants-grpab)
- [`pPrimaryInvariants(A, p): GrpAb, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/invariant.md#function-pprimaryinvariants-grpab-rngintelt)
- [Canonical Decomposition](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/decomposition.md)
- [`TorsionFreeSubgroup(A): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/decomposition.md#function-torsionfreesubgroup-grpab)
- [`TorsionSubgroup(A): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/decomposition.md#function-torsionsubgroup-grpab)
- [`pPrimaryComponent(A, p): GrpAb, RngIntElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/decomposition.md#function-pprimarycomponent-grpab-rngintelt)
- [Set-Theoretic Operations](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md)
- [Functions Relating to Group Order](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#functions-relating-to-group-order)
- [`Order(G): GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-order-grpab)
- [`# G: GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-operation-grpab-rngintelt)
- [`FactoredOrder(G): GrpAb → []`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-factoredorder-grpab)
- [`Exponent(G): GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-exponent-grpab)
- [`IsFinite(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-isfinite-grpab)
- [`IsInfinite(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-isinfinite-grpab)
- [Membership and Equality](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#membership-and-equality)
- [`g in G: GrpAbElt, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-op-in-grpabelt-grpab)
- [`g notin G: GrpAbElt, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-op-notin-grpabelt-grpab)
- [`S subset G: { GrpAbElt}, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-op-subset-grpabelt-grpab)
- [`S notsubset G: { GrpAbElt}, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-operation-notsubset-grpabelt-grpab-boolelt)
- [`H subset G: GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-op-subset-grpab-grpab)
- [`IsSubgroup(H,G): GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-issubgroup-grpab-grpab)
- [`H notsubset G: GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-operation-notsubset-grpab-grpab-boolelt)
- [`G eq H: GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-op-eq-grpab-grpab)
- [`G ne H: GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#operation-op-ne-grpab-grpab)
- [Set Operations](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#set-operations)
- [`NumberingMap(G): GrpAb → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-numberingmap-grpab)
- [`RandomProcess(G): GrpAb → Process`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-randomprocess-grpab)
- [`Random(P): Process → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-random-process)
- [`Random(G): GrpAb → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-random-grpab)
- [`Random(G, n): GrpAb, RngIntElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-random-grpab-rngintelt)
- [`Rep(G): GrpAb → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-rep-grpab)
- [`Representative(G): GrpAb → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/set.md#function-representative-grpab)
- [Coset Spaces](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md)
- [`Transversal(G, H): GrpAb, GrpAb → {@ GrpAbElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#function-transversal-grpab-grpab)
- [`RightTransversal(G, H): GrpAb, GrpAb → {@ GrpAbElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#function-righttransversal-grpab-grpab)
- [Coercions Between Groups and Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#coercions-between-groups-and-subgroups)
- [`G ! g: GrpAb, GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#operation-op-grpab-grpabelt)
- [`H ! g: GrpAb, GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#operation-op-grpab-grpabelt-2)
- [`K ! g: GrpAb, GrpAbElt → GrpAbElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#operation-op-grpab-grpabelt-3)
- [`Morphism(H, G): GrpAb, GrpAb → ModMatRngElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/coset-space.md#function-morphism-grpab-grpab)
- [Subgroup Constructions](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md)
- [`H meet K: GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#operation-op-meet-grpab-grpab)
- [`H meet:= K: GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#operation-operation-meet-grpab-grpab-grpab)
- [`H + K: GrpAb, GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#operation-op-plus-grpab-grpab)
- [`n * G: RngIntElt, GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#operation-op-times-rngintelt-grpab)
- [`FrattiniSubgroup(G): GrpAb → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#function-frattinisubgroup-grpab)
- [`SylowSubgroup(G, p : parameters): GrpAb, RngIntElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#function-sylowsubgroup-grpab-rngintelt)
- [`Sylow(G, p : parameters): GrpAb, RngIntElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#function-sylow-grpab-rngintelt)
- [Example: Sylow 2-Subgroup of a Unit Group](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-constructions.md#example-ex-c74c85)
- [Subgroup Chains](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md)
- [`ChiefSeries(G): GrpAb → [GrpAb]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-chiefseries-grpab)
- [`CompositionSeries(G): GrpAb → [GrpAb]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-compositionseries-grpab)
- [`Agemo(G, i): GrpAb, RngIntElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-agemo-grpab-rngintelt)
- [`Omega(G, i): GrpAb, RngIntElt → GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-omega-grpab-rngintelt)
- [`ElementaryAbelianSeries(G): GrpAb → [ GrpAb ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-elementaryabelianseries-grpab)
- [`DerivedSeries(G): GrpAb → [ GrpAb ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-derivedseries-grpab)
- [`LowerCentralSeries(G): GrpAb → [ GrpAb ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-lowercentralseries-grpab)
- [`UpperCentralSeries(G): GrpAb → [ GrpAb ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-uppercentralseries-grpab)
- [`SubnormalSeries(G, H): GrpAb, GrpAb → [ GrpAb ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/subgroup-chains.md#function-subnormalseries-grpab-grpab)
- [General Group Properties](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md)
- [`IsCyclic(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-iscyclic-grpab)
- [`IsElementaryAbelian(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-iselementaryabelian-grpab)
- [`IsFree(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-isfree-grpab)
- [`IsMixed(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-ismixed-grpab)
- [`IspGroup(G): GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-ispgroup-grpab)
- [`DerivedLength(G): GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-derivedlength-grpab)
- [Properties of Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#properties-of-subgroups)
- [`IsMaximal(G, H): GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-ismaximal-grpab-grpab)
- [`Index(G, H): GrpAb, GrpAb → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-index-grpab-grpab)
- [`FactoredIndex(G, H): GrpAb, GrpAb → []`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-factoredindex-grpab-grpab)
- [`IsPure(G, H): GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-ispure-grpab-grpab)
- [`IsNeat(G, H): GrpAb, GrpAb → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-isneat-grpab-grpab)
- [Enumeration of Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#enumeration-of-subgroups)
- [`MaximalSubgroups(G): GrpAb → [GrpAb]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-maximalsubgroups-grpab)
- [`Subgroups(G:parameters): GrpAb → [Rec]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-subgroups-grpab)
- [`NumberOfSubgroupsAbelianPGroup(A): SeqEnum → SeqEnum`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-numberofsubgroupsabelianpgroup-seqenum)
- [`HasComplement(G, U): GrpAb, GrpAb → BoolElt, GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#function-hascomplement-grpab-grpab)
- [Example: Subgroups of an Abelian Group of Order 12](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/group-Boolean.md#example-ex-604d09)
- [Representation Theory](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/character-representation.md)
- [`CharacterTable(G): GrpAb → TabChtr`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/character-representation.md#function-charactertable-grpab)
- [The Hom Functor](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md)
- [`Hom(G, H): GrpPC, GrpPC → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#function-hom-grppc-grppc)
- [`Hom(G, H): GrpAb, GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#function-hom-grpab-grpab)
- [`HomGenerators(G, H): GrpAb, GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#function-homgenerators-grpab-grpab)
- [`AllHomomorphisms(G, H): GrpAb, GrpAb → [Map]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#function-allhomomorphisms-grpab-grpab)
- [`AllHomomorphisms(G, H): GrpPC, GrpPC → [Map]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#function-allhomomorphisms-grppc-grppc)
- [`Homomorphisms(G, H): GrpAb, GrpAb → [Map]`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#function-homomorphisms-grpab-grpab)
- [Example: Hom Functor on Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/hom.md#example-ex-0d5388)
- [Automorphism Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/autgp.md)
- [`AutomorphismGroup(G): GrpAb → GrpAuto`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/autgp.md#function-automorphismgroup-grpab)
- [`IsIsomorphic(G, H): GrpAb, GrpAb → BoolElt, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/autgp.md#function-isisomorphic-grpab-grpab)
- [Cohomology](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/cohomology.md)
- [`Dual(G): GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/cohomology.md#function-dual-grpab)
- [`H2_G_QmodZ(G): GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/cohomology.md#function-h2-g-qmodz-grpab)
- [`Res_H2_G_QmodZ(U, H2): GrpAb, GrpAb → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/cohomology.md#function-res-h2-g-qmodz-grpab-grpab)
- [Homomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/homomorphisms.md)
- [`hom< A -> B | L>: Grp, Grp, List → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/homomorphisms.md#constructor-constructor-hom-grp-grp-list-map)
- [`Homomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/homomorphisms.md#function-homomorphism-grp-grp-grpelt-grpelt)
- [`iso< A -> B | L>: Grp, Grp, List → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/homomorphisms.md#constructor-constructor-iso-grp-grp-list-map)
- [`Isomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/homomorphisms.md#function-isomorphism-grp-grp-grpelt-grpelt)
- [Example: Homomorphisms of Generic Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/homomorphisms.md#example-ex-32bea9)
- [Bibliography](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/bibliography.md)
### Introduction
This Chapter contains a description of the Magma machinery provided for computing with abstract abelian groups. Such a group may be described either in terms of defining relations (category `GrpAb`) or by defining the group operations on some finite set (category `GrpAbGen`). In the case of finitely presented groups, the abelian groups may be finite and infinite, the only restriction being that the group be finitely generated.
### Construction of a Finitely Presented Abelian Group and its Elements
#### The Free Abelian Group
##### `FreeAbelianGroup(n): RngIntElt -> GrpAb`
Construct the free abelian group $F$ on $n$ generators, where $n$ is a positive integer. The $i$-th generator may be referenced by the expression `F.i`, $i = 1, \ldots, n$. Note that a special form of the assignment statement is provided which enables the user to assign names to the generators of $F$. In this form of assignment, the list of generator names is enclosed within angle brackets and appended to the variable name on the *left hand side* of the assignment statement.
##### `Example: Free Abelian Group (ex-c01eb3)`
The statement
```magma
> F := FreeAbelianGroup(2);
```
creates the free abelian group on two generators. Here the generators may be referenced using the standard names, $F.1$ and $F.2$.
The statement
```magma
> F := FreeAbelianGroup(2);
```
defines $F$ to be the free abelian group on two generators and assigns the names $x$ and $y$ to the generators.
#### Relations
##### `w₁ = w₂: GrpAbElt, GrpAbElt -> Rel`
Given words $w_1$ and $w_2$ over the generators of an abelian group $A$, create the relation $w_1 = w_2$. Note that this relation is not automatically added to the existing set of defining relations $R$ for $S$. It may be added to $R$, for example, through use of the `quo`-constructor (see below).
##### `r[1]: GrpAbRel, RngIntElt -> GrpAbElt`
##### `LHS(r): Rel -> GrpAbElt`
Given a relation $r$ over the generators of $A$, return the left hand side of the relation $r$. The object returned is a word over the generators of $A$.
##### `r[2]: GrpAbRel, RngIntElt -> GrpAbElt`
##### `RHS(r): Rel -> GrpAbElt`
Given a relation $r$ over the generators of $A$, return the right hand side of the relation $r$. The object returned is a word over the generators of $A$.
##### `Parent(r): RelElt -> GrpAb`
Group over which the relation $r$ is taken.
##### `Example: Relations in a Free Abelian Group (ex-0f1824)`
We may define a group and a set of relations as follows:
```magma
> F := FreeAbelianGroup(2);
> rels := { 2*x = 3*y, 4*x + 4*y = Id(F)} ;
```
To replace one side of a relation, the easiest way is to reassign the relation. So for example, to replace the relation $2 x = 3 y$ by $2 x = 4 y$:
```magma
> r := 2*x = 3*y;
> r := LHS(r) = 4*y;
```
#### Specification of a Presentation
An abelian group with non-trivial relations is constructed as a quotient of an existing abelian group, possibly a free abelian group.
##### `AbelianGroup< X | R >: List(Var), List(GrpAbRel) -> GrpAb, Hom(GrpAb)`
Given a list $X$ of variables $x_1, \ldots, x_r$, and a list of relations $R$ over these generators, let $F$ be the free abelian group on the generators $x_1, \ldots, x_r$. Then construct (a) an abelian group $A$ isomorphic to the quotient of $F$ by the subgroup of $F$ defined by $R$, and (b) the natural homomorphism $\phi : F \rightarrow A$.
Each term of the list $R$ is either a *word*, a *relation*, a *relation list*, or a *subgroup* of $F$.
- A *relation* consists of a pair of words, separated by ‘=’.
- A *word* $w$ is interpreted as a *relator*, that is, it is equivalent to the relation $w = 0$. (See above).
- A *relation list* consists of a list of words, where each pair of adjacent words is separated by ‘=’: $w_1 = w_2 = \cdots = w_r$. This is interpreted as the set of relations $w_1 = w_r, \ldots, w_{r - 1} = w_r$. Note that the relation list construct is only meaningful in the context of the `quo`-constructor.
A *subgroup* $H$ appearing in the list $R$ contributes its generators to the relation set for $A$, i.e., each generator of $H$ is interpreted as a relator for $A$. The group $F$ may be referred to by the special symbol `$` in any word appearing to the right of the ‘`|`’ symbol in the `quo`-constructor. Also, in the context of the `quo`-constructor, the identity element (empty word) may be represented by the digit $0$.
The function returns:
**(a)**
The quotient group $A$;
**(b)**
The natural homomorphism $\phi : F \rightarrow A$.
##### `Example: Abelian Group from a Presentation (ex-bd5820)`
We create the abelian group defined by the presentation $< a, b, c\ |\ 7a + 4b + c, 8a + 5b + 2c, 9a + 6b + 3c >$.
```magma
> F := FreeAbelianGroup(3);
> A := quo< F | 7*a + 4*b + c, 8*a + 5*b + 2*c, 9*a + 6*b + 3*c >;
> A;
AbelianGroup isomorphic to Z_3 + Z
Defined on 2 generators
Relations:
3*A.1 = 0
```
A simple way of specifying an abelian group is as a product of cyclic groups.
##### `AbelianGroup([n₁,...,nᵣ]): [ RngIntElt ] -> GrpAb`
Construct the abelian group defined by the sequence $[n_1,\ldots,n_r]$ of non-negative integers as an abelian group. The function returns the direct product of cyclic groups $C_{n_1} \times C_{n_2} \times \cdots \times C_{n_r}$, where $C_0$ is interpreted as an infinite cyclic group.
##### `Example: Abelian Group from Cyclic Factors (ex-ea4c01)`
We create the abelian group $Z_2 \times Z_3 \times Z_4 \times Z_5 \times Z_6 \times Z \times Z$.
```magma
> G<[x]> := AbelianGroup([2,3,4,5,6,0,0]);
> G;
Abelian Group isomorphic to Z/2 + Z/6 + Z/60 + Z + Z
Defined on 7 generators
Relations:
2*G.1 = 0
3*G.2 = 0
4*G.3 = 0
5*G.4 = 0
6*G.5 = 0
```
#### Accessing the Defining Generators and Relations
The functions described here provide access to basic information stored for an abelian group $A$.
##### `A . i: GrpAb, RngIntElt -> GrpAbElt`
The $i$-th defining generator for $A$.
##### `Generators(A): GrpAb -> { GrpAbElt}`
A set containing the generators for $A$.
##### `NumberOfGenerators(A): GrpAb -> RngIntElt`
##### `Ngens(A): GrpAb -> RngIntElt`
The number of generators for $A$.
##### `Parent(u): GrpAbElt -> GrpAb`
The parent group $A$ of the word $u$.
##### `Relations(A): GrpAb -> [ Rel ]`
A sequence containing the defining relations for $A$.
##### `RelationMatrix(A): GrpAb -> Mtrx`
A matrix where each row corresponds to one of the defining relations of $A$.
### Construction of a Generic Abelian Group
The term generic abelian group refers to the situation in which a group $A$ is described by giving a set $X$ together with functions acting on $X$ that perform the fundamental group operations for $A$. Specifically, the user must provide functions which compute the product, the inverse, and the identity. For efficiency, the user may also optionally specify the order and a generating set for the group. This is made explicit below.
Going from such a definition of an abelian group $A$ to a presentation for $A$ will often be extremely expensive and so a small number of operations are implemented so as to not require this information. The two key operations are computing the order of an element and computing the discrete logarithm of an element to a given base. For many abelian group operations, knowledge of a presentation is required and if such an operation is invoked, the structure of $A$ (and hence a presentation) will be automatically constructed.
There are two possible ways of computing the structure of the group. If the order of the group is known (or can be computed) then it is relatively easy to construct each of the $p$-Sylow subgroups. If the order is not available, the structure is computed from a set of generators supplied by the user.
#### Specification of a Generic Abelian Group
##### `GenericAbelianGroup(U: parameters): . -> GrpAbGen`
```magma
IdIntrinsic : MonStg Default:
AddIntrinsic : MonStg Default:
InverseIntrinsic : MonStg Default:
UseRepresentation: Bool Default: false
Order : RngIntElt Default:
UserGenerators : SetEnum Default:
ProperSubset : Bool Default: false
RandomIntrinsic : MonStg Default:
ComputeStructure : Bool Default: false
UseUserGenerators: Bool Default: false
PollardRhoRParam : RngInt Default: 20
PollardRhoTParam : RngInt Default: 8
PollardRhoVParam : RngInt Default: 3
```
Construct the generic abelian group $A$ over the domain $U$. The domain $U$ can be an aggregate (set, indexed set, or sequence) of elements or it can be any structure, for example, an elliptic curve, a jacobian, a magma of quadratic forms.
If the parameters `IdIntrinsic`, `AddIntrinsic`, and `InverseIntrinsic` are not set, then the identity, the composition, and the inverse function of $A$ are taken to be the identity, the composition, and the inverse function of $U$ or of `Universe(U)` if $U$ is an aggregate. If the parameters `IdIntrinsic`, `AddIntrinsic`, and `InverseIntrinsic` are set, they define the identity, the composition, and the inverse function of $A$.
The parameter `IdIntrinsic` must be a function name whose sole argument is $U$ or `Universe(U)` if $U$ is an aggregate. `AddIntrinsic` must be a function name whose only two arguments are elements of $U$ or of `Universe(U)` if $U$ is an aggregate. `InverseIntrinsic` must be a function name whose only two arguments are elements of $U$ or of `Universe(U)` if $U$ is an aggregate. That is, it is required that `InverseIntrinsic` be a binary operation (see the example below where `InverseIntrinsic := "/"` is indeed binary). Defining any of the three above parameters implies that the other two must be defined as well.
Setting the parameter `UseRepresentation` to true implies that all elements of $A$ will be internally represented as linear combinations of the generators of $A$ obtained while computing the structure of $A$. This can be a costly procedure, since such a representation is akin to solving the discrete logarithm problem. The advantage of such a representation is that arithmetic for elements of $A$ as well as computing the order of elements of $A$ are then essentially trivial operations.
The parameter `Order` can be set to the order of the group, if known. Knowledge of the order may save a considerable amount of time for operations such as computing a Sylow subgroup, determining the group structure, or solve a discrete logarithm problem. More importantly, if $A$ is a proper subset of $U$, or of `Universe(U)` if $U$ is an aggregate, then one of `Order` or `UserGenerators` must be set.
One can set `UserGenerators` to some set of elements of $U$, or of `Universe(U)` if $U$ is an aggregate, which generate the group $A$. This can be useful when $A$ is a subset of $U$ (`Universe(U)`), or more generally when the computation of the group structure of $A$ is made from a set of generators. Finally, setting `UserGenerators` may be an alternative to setting `RandomIntrinsic`.
The parameter `ProperSubset` must be set when $A$ is a proper subset of $U$ (`Universe(U)`).
The parameter `RandomIntrinsic` indicates the random function to use. If it is not set, the random function (if it is required) is taken to be the random function applying to $U$ (`Universe(U)`).
The parameter `RandomIntrinsic` must be the name of a function taking as its sole argument $U$ (`Universe(U)`) and returning an element of $U$ (`Universe(U)`) which is also an element of the group $A$, which is important when $A$ is a proper subset of $U$ (`Universe(U)`. Therefore, if $A$ is a proper subset of $U$ (`Universe(U)`), then `RandomIntrinsic` must be set, unless the parameter `UserGenerators` is set.
The parameter `Structure` indicates that the group structure should be determined at the time of creation. If this parameter is set then the user may also want to set the following parameters:
Their meaning is detailed in Section [Computing Abelian Group Structure](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/generic-abelian.md#grpab-sec-structure).
##### `Example: Generic Abelian Group of Units (ex-5df08f)`
In our first example we create the subset $U$ of ${\mathbb{Z}}/34384{\mathbb{Z}}$ corresponding to its unit group as a generic abelian group $G$. Note that $U$ is an indexed set.
```magma
> m := 34384;
> Zm := Integers(m);
> U := {@ x : x in Zm | GCD(x, m) eq 1 @};
> G := GenericAbelianGroup(U :
> IdIntrinsic := "Id",
> AddIntrinsic := "*",
> InverseIntrinsic := "/");
> G;
Generic Abelian Group over
Residue class ring of integers modulo 34384
```
In our next example we construct unique representatives for the classes of binary quadratic forms corresponding to elements of a subgroup of the class group of the imaginary quadratic field of discriminant $-4000004$.
```magma
> n := 6;
> Dk := -4*(10^n + 1);
> Q := QuadraticForms(Dk);
>
> p := 2;
> S := { };
> while #S lt 10 do
> if KroneckerSymbol(Dk,p) eq 1 then
> I := Reduction(PrimeForm(Q,p));
> Include(~S, I);
> end if;
> p := NextPrime(p);
> end while;
>
> QF := GenericAbelianGroup(Q : UserGenerators := S,
> ComputeStructure := true,
> UseUserGenerators := true);
> QF;
Generic Abelian Group over
Binary quadratic forms of discriminant -4000004
Abelian Group isomorphic to Z/2 + Z/516
Defined on 2 generators
Relations:
2*$.1 = 0
516*$.2 = 0
```
So the structure of the subgroup is $Z_2 \times Z_{516}$
#### Accessing Generators
These functions give access to generating sets for a generic group $A$. If a generating set is requested and none has been supplied by the user then this will require the determination of the group structure which could be very expensive. Note the distinction between `Generators` and `UserGenerators`. From now on, unless otherwise specified, whenever mention is made of the generators of $A$, we refer to the generators as given by the `Generators` function.
##### `Universe(A): GrpAbGen`
The universe over which the generic abelian group $A$ is defined.
##### `A . i: GrpAbGen, RngIntElt -> GrpAbGenElt`
The $i$-th generator for the generic abelian group $A$.
##### `Generators(A): GrpAbGen -> [ GrpAbGenElt ]`
A sequence containing a generating set for the generic abelian group $A$ as elements of $A$. The set of generators returned for $A$ is a reduced set of generators as constructed during the structure computation. Therefore, if the group structure of $A$ has been computed from a user-supplied set of generators, it is unlikely that `Generators(A)` and `UserGenerators(A)` will be the same.
##### `UserGenerators(A): GrpAbGen -> [ GrpAbGenElt ]`
A sequence containing the user-supplied generators for the generic abelian group $A$ as elements of $A$.
##### `NumberOfGenerators(A): GrpAbGen -> RngIntElt`
##### `Ngens(A): GrpAbGen -> RngIntElt`
The number of generators for the generic abelian group $A$.
#### Computing Abelian Group Structure
If the order of a generic abelian group $A$ can be obtained then the structure of $A$ is found by constructing each $p$-Sylow subgroup of $A$. The $p$-Sylow subgroups are built from random elements of the underlying set $X$ of $A$. Recall that $U$ (`Universe(U)`) is the domain of $A$. Random elements are obtained using either a random function attached to $X$ or using a user-supplied function (the `RandomIntrinsic` parameter), or using user-supplied generators (the `UserGenerators` parameter). It is therefore vital that user-supplied generators truly generate the group one wishes to construct. A drawback of this method of obtaining the structure of $A$ is the memory needed to store all the elements of a specific $p$-Sylow subgroup while under construction. This algorithm is mostly based on work by Michael Stoll.
The second approach computes the group structure from a set of generators as supplied by the user, removing the need to compute the order of $A$. This can be particularly useful when computing this order is expensive. Note that computing the structure of a group from a set of generators is akin to solving a number of the discrete logarithm problems. This second algorithm uses a variant of the Pollard–Rho algorithm and is due to Edlyn Teske [[Tes98a](https://docs.magma-maths.org/references.md#cite-teske-prho)].
If $A$ is a subgroup of a generic abelian group, $G$ say, then the structure of $G$ is computed first. The rationale is that once the structure of $G$ is known, computing the structure of $A$ is almost always cheap.
##### `AbelianGroup(A: parameters): GrpAbGen -> GrpAb, Map`
```magma
UseUserGenerators: Bool Default: false
PollardRhoRParam : RngInt Default: 20
PollardRhoTParam : RngInt Default: 8
PollardRhoVParam : RngInt Default: 3
```
Compute the group structure of the generic abelian group $A$, which may be a subgroup as created by the subgroup constructor or the Sylow function. The two values returned are the abstract abelian group and the invertible map from the latter into $A$.
If `UseUserGenerators` is false, then the group structure computation is made via the construction of each $p$-Sylow subgroup, using the factorization of the order of $A$.
If `UseUserGenerators` is set to `true`, the group structure computation uses the user-supplied set of generators for $A$. In this case, the additional parameters `PollardRhoRParam`, `PollardRhoTParam`, and `PollardRhoVParam` can be supplied. Their values will be passed to the Pollard–Rho algorithm used in the group structure computation: `PollardRhoRParam` sets the size of the $r$-adding walks, `PollardRhoTParam` sets the size of the internal storage of elements, and `PollardRhoVParam` is used for an efficient finding of the periodic segment. It is conjectured that the default values as given above are ‘best’ (see [[Tes98b](https://docs.magma-maths.org/references.md#cite-teske-walks)]), therefore there should be no need to set these parameters in general.
##### `Example: Structure of a Unit Group (ex-0d76ee)`
The following statement computes the structure of the unit group of ${\mathbb{Z}}_{34384}$.
```magma
> G := AbelianGroup(G); G;
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/2 + Z/2 + Z/6 + Z/612
Defined on 4 generators
Relations:
2*G.1 = 0
2*G.2 = 0
6*G.3 = 0
612*G.4 = 0
```
### Elements
#### Construction of Elements
Unless otherwise stated, the operations in this section apply to fp-abelian groups and generic abelian groups.
##### `A ! [a₁, ... ,aₙ]: GrpAb, [RngIntElt] -> GrpAbElt`
##### `A ! [a₁, ... ,aₙ]: GrpAbGen, [RngIntElt] -> GrpAbGenElt`
Given an abelian group $A$ with generators $e_1, \ldots, e_r$ and a sequence $Q = [a_1, \cdots, a_r]$ of integers, construct the element $a_1 e_1 + \cdots + a_r e_r$ of $A$.
##### `A ! e: GrpAbGen, Elt -> GrpAbGenElt`
Given a generic abelian group $A$ and an element $e$ of the domain over which it is defined, return $e$ as an element of $A$. If $A$ is a proper subset of its underlying domain, then $e$ must be a linear combination of the generators (which may be user-supplied) of $A$.
##### `A ! g: GrpAbGen, GrpAbGenElt -> GrpAbGenElt`
Given a generic abelian group $A$ and an element $g$ of the underlying set $X$ of $A$, return $g$ as an element of $A$.
##### `A ! n: GrpAb, RngIntElt -> GrpAbElt`
Given an abelian group $A$ with exactly one generator $x$, construct the element $n x$.
##### `Random(A): GrpAbGen -> GrpAbGenElt`
Given either a finite fp-abelian group or a generic abelian group $A$, return a random element of $A$.
##### `Identity(A): GrpAb -> GrpAbElt`
##### `Id(A): GrpAb -> GrpAbElt`
##### `A ! 0: GrpAb, RngIntElt -> GrpAbElt`
Construct the identity element (empty word) for the abelian group $A$.
Let $A$ be a generic abelian group defined in the universe $U$ of $A$. If $g$ is an element of $A$, then $U!g$ is an element of $U$.
#### Representation of an Element
An element $g$ of an abelian group $A$ can be represented as a linear combination with respect to a given generating sequence. The coefficients appearing in this linear combination provide an alternative representation for $g$. If $A$ is a fp-group, the generating set will be the one on which the group was defined. In the case of a generic group, the generating set can either be that obtained when constructing a presentation for $A$ or a user-supplied generating set.
##### `Representation(g): GrpAbGenElt -> [RngIntElt]`
##### `ElementToSequence(g): GrpAbGenElt -> [RngIntElt]`
##### `Eltseq(g): GrpAbGenElt -> [RngIntElt]`
Let $A$ be an abelian group with generating set $e_1, \ldots, e_n$ and suppose $g$ is an element of $A$, where $g = a_1 e_1 + \ldots + a_n e_n$. These functions return the sequence $Q$ of $n$ integers defined by $Q[i] = {a_i}$, for $i = 1,\ldots,n$. Moreover, each $a_i$, $i = 1,\ldots,n$, is the integer residue modulus the order of the $i$th generator.
##### `UserRepresentation(g): GrpAbGenElt -> [RngIntElt]`
Let $A$ be a generic abelian group with a user-supplied set of generators $u_1, \ldots, u_n$ and suppose $g$ is an element of $A$, where $g = a_1 u_1 + \ldots + a_n u_n$. This function returns the sequence $Q$ of $n$ integers defined by $Q[i] = {a_i}$, for $i = 1,\ldots,n$. Moreover, each $a_i$, $i = 1,\ldots,n$, is the integer residue modulus the order of the $i$th generator.
##### `Representation(S, g): SeqEnum, GrpAbGenElt -> [RngIntElt], RngIntElt`
Let $A$ be a generic abelian group and let $S = [s_1, \dots, s_m]$ be any sequence of elements of $A$. Assume $g$ is an element of $A$ such that $b g = a_1 s_1 + \ldots + a_m s_m$. This function returns as its first value the sequence $Q$ of $m$ integers defined by $Q[i] = {a_i}$, for $i = 1,\ldots,m$. The second value returned is the coefficient $b$ of $g$. Note that $b$ might not be 1.
##### `Example: Creating and Representing Elements (ex-f86c65)`
We use the quadratic forms example considered above to illustrate these functions.
```magma
> Generators(QF);
[ <2,2,500001>, <206,-102,4867> ]
> g := QF ! [5, 6];
> g;
<837,-766,1370>
> Representation(g);
[ 1, 6 ]
>
> g := Random(QF);
> Representation(g);
[ 1, 270 ]
>
> UserRepresentation(g);
[ 377, 0, 515, 0, 0, 0, 0, 0, 0, 0 ]
>
> S := [];
> for i in [1..3] do
> d := Random(QF);
> Include(~S, d);
> end for;
> seq, coeff := Representation(S, g);
> seq; coeff;
[ -170, -3, 0 ]
1
```
#### Arithmetic with Elements
If the generic abelian group $A$ has been constructed with the flag `UseRepresentation` set true, then arithmetic with elements of $A$ is trivial.
##### `u + v: GrpAbElt, GrpAbElt -> GrpAbElt`
Given elements $u$ and $v$ belonging to the same abelian group $A$, return the sum of $u$ and $v$.
##### `- u: GrpAbElt -> GrpAbElt`
The inverse of element $u$.
##### `u - v: GrpAbElt, GrpAbElt -> GrpAbElt`
Given elements $u$ and $v$ belonging to the same abelian group $A$, return the sum of $u$ and the inverse of $v$.
##### `m * u: RngIntElt, GrpAbElt -> GrpAbElt`
##### `u * m: GrpAb, RngIntElt -> GrpAbElt`
Given an integer $m$, return the element $w+w+\cdots w$ ($|m|$ summands), where $w = u$, if $m$ is positive and $w = -u$ if $m$ is negative.
### Construction of Subgroups and Quotient Groups
The operations in this section apply to both, free abelian groups and arbitrary abelian groups.
#### Construction of Subgroups
##### `sub: GrpAb, List -> GrpAb, Map`
Construct the subgroup $B$ of the abelian group $A$ generated by the elements specified by the terms of the *generator list* $L$. A term $L[i]$ of the generator list may consist of any of the following objects:
**(a)**
An element liftable to A;
**(b)**
A sequence of integers representing an element of A;
**(c)**
A subgroup of $A$;
**(d)**
A set or sequence of type (a), (b), or (c).
The collection of words and groups specified by the list must all belong to the group $A$ and the group $B$ will be constructed as a subgroup of $A$.
##### `Example: Creating a Subgroup of an Abelian Group (ex-f256c6)`
We create a subgroup of the group $A = Z_2 + Z_3 + Z_4 + Z_5 + Z_6 + Z + Z$.
```magma
> A<[x]> := AbelianGroup([2,3,4,5,6,0,0]);
> A;
Abelian Group isomorphic to Z/2 + Z/6 + Z/60 + Z + Z
Defined on 7 generators
Relations:
2*x[1] = 0
3*x[2] = 0
4*x[3] = 0
5*x[4] = 0
6*x[5] = 0
> B<[y]> := sub< A | x[1], x[3], x[5], x[7] >;
> B;
Abelian Group isomorphic to Z/2 + Z/2 + Z/12 + Z
Defined on 4 generators in supergroup A:
y[1] = 2*x[3] + 3*x[5]
y[2] = x[1]
y[3] = 3*x[3] + x[5]
y[4] = x[7]
Relations:
2*y[1] = 0
2*y[2] = 0
12*y[3] = 0
```
In the case of subgroups of generic groups, a number of parameters are provided.
##### `sub: GrpAbGen, List -> GrpAbGen`
```magma
Order : RngInt Default:
RandomIntrinsic : MonStg Default:
ComputeStructure : Bool Default: false
UseUserGenerators: Bool Default: false
PollardRhoRParam : RngInt Default: 20
PollardRhoTParam : RngInt Default: 8
PollardRhoVParam : RngInt Default: 3
```
Construct the subgroup of the generic abelian group $A$ generated by the elements specified by the terms of the *generator list* $L$. A term $L[i]$ of the generator list may consist of any of the following objects:
**(a)**
An element liftable into $A$;
**(b)**
A sequence of integers representing an element of $A$;
**(c)**
A set or sequence whose elements may be of either of the above types.
An element liftable into $A$ may be an element of $A$ itself, or it may be an element of $U$ (`Universe(U)`), $U$ being as usual the domain over which $A$ is defined. For consistency with the construction, the values of the following parameters may also be passed to the subgroup constructor:
In particular, it is possible to construct a subgroup by giving its order and a random function generating elements of the *subgroup*. In this case, the list $L$ would be empty since calculation of the subgroup structure would result in the construction of the $p$-Sylow subgroups from random elements of the subgroup. Further, when the structure of $A$ is already known and if the subgroup is defined in terms of a set of generators in $L$ then the subgroup structure is computed at the time of creation.
##### `Example: Subgroup of a Quadratic Forms Group (ex-8e5822)`
We create a subgroup of the quadratic forms group considered above.
```magma
> S := [];
> for j in [1..2] do
> P := Random(QF);
> Include(~S, P);
> end for;
> S;
[ <45,26,22226>, <937,-930,1298> ]
> QF1 := sub< QF | S>;
> QF1;
Generic Abelian Group over
Binary quadratic forms of discriminant -4000004
Abelian Group isomorphic to Z/2 + Z/258
Defined on 2 generators in supergroup A:
QF1.1 = QF.1
QF1.2 = 2*QF.2
Relations:
2*QF1.1 = 0
258*QF1.2 = 0
>
```
#### Construction of Quotient Groups
##### `quo: GrpAb, List -> GrpAb, Hom(GrpAb)`
Given an abelian group $F$, and a set of relations $R$ in the generators of $F$, construct (a) an abelian group $A$ isomorphic to the quotient of $F$ by the subgroup of $F$ defined by $R$, and (b) the natural homomorphism $\phi : F \rightarrow A$.
The expression defining $F$ may be either simply the name of a previously constructed group, or an expression defining an abelian group. The possibilities for the relation list $R$ are the same as for the `AbelianGroup` construction.
The function returns:
**(a)**
The quotient group $A$;
**(b)**
The natural homomorphism $\phi : F \rightarrow A$.
##### `A / B: GrpAb, GrpAb -> GrpAb`
Given a subgroup $B$ of the abelian group $A$, construct the quotient of $A$ by $B$.
### Standard Constructions and Conversions
#### `AbelianGroup(GrpAb, Q): Cat, [ RngIntElt ] -> GrpAb`
#### `AbelianGroup(Q): [ RngIntElt ] -> GrpAb`
Let $Q = [ a_1, \ldots, a_r]$ be a sequence of non-negative integers. This function creates the abelian group ${\mathbb{Z}}_1 + \cdots + {\mathbb{Z}}_r$, where $Z_i$ is the cyclic group of order $|a_i|$ if $a_i\neq0$ or the infinite cyclic group ${\mathbb{Z}}$ otherwise, $i = 1, \ldots, r$.
#### `AbelianGroup(G): Grp -> GrpAb, Hom`
Given an abelian permutation, matrix, or polycyclic group $G$, represent it as an abelian group $A$. The function also returns the isomorphism $\phi: G \ar A$ as its second value.
#### `AbelianQuotient(G): Grp -> GrpAb, Hom`
Given a finitely presented, permutation, matrix, or polycyclic group $G$, return the maximal abelian quotient $A$ of $G$. The function returns the natural homomorphism $\phi: G \ar A$ as its second value.
#### `DirectSum(A, B): GrpAb, GrpAb -> GrpAb`
The direct sum of abelian groups $A$ and $B$.
#### `PCGroup(A): GrpAb -> GrpPC, Hom(Grp)`
A pc-group representation $G$ of $A$. The isomorphism $\phi: A\rightarrow G$ is also returned.
#### `PermutationGroup(A): GrpAb -> GrpPerm, Hom(Grp)`
A permutation group representation of $A$. The particular group $G$ is generated by disjoint cycles whose lengths are the abelian invariants of $A$. The isomorphism $\phi: G\rightarrow A$ is also returned.
#### `FPGroup(A): GrpAb -> GrpFP, Hom(Grp)`
A fp-group representation of $A$. The particular group $G$ is generated by commuting generators whose orders are the abelian invariants of $A$. The isomorphism $\phi: G\ar A$ is also returned.
#### `CommutatorSubgroup(G): GrpAb -> GrpAb`
#### `DerivedSubgroup(G): GrpAb -> GrpAb`
#### `DerivedGroup(G): GrpAb -> GrpAb`
The derived subgroup of $G$, that is the trivial group, since $G$ is abelian.
#### `CommutatorSubgroup(H, K): GrpAb, GrpAb -> GrpAb`
#### `CommutatorSubgroup(G, H, K): GrpAb, GrpAb, GrpAb -> GrpAb`
The commutator subgroup of groups $H$ and $K$ in their common overgroup $G$.
#### `Centralizer(G, a): GrpAb, GrpAbElt -> GrpAb`
#### `Centraliser(G, a): GrpAb, GrpAbElt -> GrpAb`
The centraliser of $a$ in $G$.
#### `Core(G, H): GrpAb, GrpAb -> GrpAb`
The maximal normal subgroup of $G$ that is contained in the subgroup $H$ of $G$. Since $G$ is abelian, this is $H$ itself.
#### `Centre(G): GrpAb -> GrpAb`
#### `Center(G): GrpAb -> GrpAb`
The center of $G$, ie. $G$ itself.
#### `FittingGroup(G): GrpAb -> GrpAb`
#### `FittingSubgroup(G): GrpAb -> GrpAb`
The Fitting subgroup of $G$.
#### `Hypercentre(G): GrpAb -> GrpAb`
#### `Hypercenter(G): GrpAb -> GrpAb`
The hypercentre of $G$.
### Operations on Elements
#### Order of an Element
##### `Order(x): GrpAbElt -> RngIntElt`
Order of the element $x$ belonging to an fp-abelian group. If the element has infinite order, the value zero is returned.
##### `Example: Orders of Abelian Group Elements (ex-5056af)`
We compute the orders of some elements in the group $Z_2 \times Z_3 \times Z_4 \times Z_5 \times Z$.
```magma
> G<[x]> := AbelianGroup([2,3,4,5,0]);
> Order( x[1] + 2*x[2] + 3*x[4]);
30
> Order( x[1] + x[5] );
0
```
It is possible to determine the order of an element of a generic group with first calculating the structure of the group. The order function involves the use of one of several algorithms:
**–**
an improved baby-step giant-step algorithm, due to J. Buchmann, M.J. Jacobson, E. Teske [[BJT97](https://docs.magma-maths.org/references.md#cite-teske-bsgs)],
**–**
the Pollard–Rho method based algorithm, described above [[Tes98a](https://docs.magma-maths.org/references.md#cite-teske-prho)].
When computing the order of an element whose lower and upper bounds are known, or where lower and upper bounds for the group order are known, the following two algorithms have been shown to be significantly faster than the two algorithms mentioned above.
**–**
the standard baby-step giant-step Shanks algorithm,
**–**
another variant of the Pollard–Rho method which is due to P. Gaudry and R. Harley [[GH](https://docs.magma-maths.org/references.md#cite-gaudry-har)].
To avoid confusion we will distinguish the algorithms due to Teske *et al* and name them the `T` baby-step giant-step algorithm and the `T` Pollard–Rho algorithm respectively. The Pollard–Rho algorithm has smaller space requirements than the baby-step giant-step algorithm, so it is recommended for use in very large groups.
In all cases, if the group order is known beforehand, the element order is computed using this knowledge. This is a trivial operation.
##### `Order(g: parameters): GrpAbGenElt -> RngIntElt`
```magma
ComputeGroupOrder: Bool Default: true
BSGSLowerBound : RngIntElt Default: 0
BSGSStepWidth : RngIntElt Default: 0
```
Assume that $g$ is an element of the generic abelian group $A$. This functions returns the order of the element $g$. Note that if `UseRepresentation` is set to true for $A$, then this is a trivial operation.
If the parameter `ComputeGroupOrder` is true, the order of $A$ is computed first (unless it is already known). The order of $g$ is then computed using this knowledge; this last computation is trivial.
If `ComputeGroupOrder` is false, the order of $g$ is computed using the `T` baby-step giant-step algorithm.
`BSGSLowerBound` and `BSGSStepWidth` are parameters which can be passed to the baby-step giant-step algorithm.
`BSGSLowerBound` sets a lowerbound on the order of the element $g$, and `BSGSStepWidth` sets the step-width in the algorithm.
##### `Order(g, l, u: parameters): GrpAbGenElt, RngIntElt, RngIntElt -> RngIntElt`
```magma
Alg : MonStg Default: "Shanks"
UseInversion: BoolElt Default: false
```
Assume that $g$ is an element of the generic abelian group $A$ such that the order of $g$ or the order of $A$ is bounded by $u$ and $l$. This function returns the order of the element $g$.
If the parameter `Alg` is set to `"Shanks"` then the generic Shanks algorithm is used, and when `Alg` is set to `"PollardRho"`, the Gaudry–Harley Pollard–Rho variant is used. Setting `UseInversion` halves the search space. To be effective element inversion must be fast.
##### `Order(g, l, u, n, m: parameters): GrpAbGenElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> RngIntElt`
```magma
Alg : MonStg Default: "Shanks"
UseInversion: BoolElt Default: false
```
Assume that $g$ is an element of the generic abelian group $A$ such that the order of $g$ or the order of $A$ is bounded by $u$ and $l$. Assume also that `Order(g)` $\equiv n$ `(mod m)` or that $\#A \equiv$ `(mod m)` This function returns the order of the element $g$. The two parameters `Alg` and `UseInversion` have the same meaning as for the previous `Order` function.
#### Discrete Logarithm
##### `Log(g, d: parameters): GrpAbGenElt, GrpAbGenElt -> RngIntElt`
```magma
ComputeGroupOrder : Bool Default: true
AlInPohligHellmanLoop: MonStg Default: "BSGS"
BSGSStepWidth : RngIntElt Default: 0
PollardRhoRParam : RngInt Default: 20
PollardRhoTParam : RngInt Default: 8
PollardRhoVParam : RngInt Default: 3
```
Assume that $g$ and $d$ are elements of the generic abelian group $A$. This function returns the discrete logarithm of $d$ to the base $g$. If `ComputeGroupOrder` is true, then the group order is computed first (if not already known) and from this the order of the base $g$ is computed. The discrete logarithm problem is then solved using a Pohlig–Hellman loop calling either the `T` baby-step giant-step algorithm (`AlInPohligHellmanLoop := "BSGS"`) or the `T` Pollard–Rho algorithm (`AlInPohligHellmanLoop := "PollardRho"`).
The parameter `BSGSStepWidth` has the same meaning as for the Order function. Parameters `PollardRhoRParam`, `PollardRhoTParam`, and `PollardRhoVParam` have the same meaning as they do for the determination of structure (function AbelianGroup). If `ComputeGroupOrder` is false then the discrete logarithm problem is solved using the `T` baby-step giant-step algorithm. Here again the parameter `BSGSStepWidth` applies.
##### `Example: Discrete Logarithm for Quadratic Forms (ex-cb963b)`
It is assumed that the structure of the groups $QF$ has already been computed. We illustrate the computation of the discrete logarithm relative to a given base.
```magma
> n := 38716;
> Ip := Reduction(PrimeForm(Q,11));
> g := GA_qf!Ip;
> d := n * g;
>
> l1 := Log(g, d : BSGSStepWidth := Floor((-Dk)^(1/4)/2));
> l2 := Log(g, d : AlInPohligHellmanLoop := "PollardRho");
> l3 := Log(g, d : ComputeGroupOrder := false);
> l4 := Log(g, d: ComputeGroupOrder := false,
> BSGSStepWidth := Floor((-Dk)^(1/4)/2));
> assert l1 eq l2 and l2 eq l3 and l3 eq l4;
> assert IsDivisibleBy(n - l1, Order(g));
```
#### Equality and Comparison
##### `u eq v: GrpAbElt, GrpAbElt -> BoolElt`
Returns `true` if the elements $u$ and $v$ are identical (as elements of the appropriate free abelian group), `false` otherwise.
##### `u ne v: GrpAbElt, GrpAbElt -> BoolElt`
Returns `true` if the elements $u$ and $v$ are not identical (as elements of the appropriate free abelian group), `false` otherwise.
##### `IsIdentity(u): GrpAbElt -> BoolElt`
##### `IsId(u): GrpAbElt -> BoolElt`
Returns `true` if the element $u$, belonging to the abelian group $A$, is the identity element (zero), `false` otherwise.
### Invariants of an Abelian Group
#### `ElementaryAbelianQuotient(G, p): GrpAb, RngIntElt -> GrpAb, Map`
The maximal $p$-elementary abelian quotient of the group $G$ as `GrpAb`. The natural epimorphism is returned as second value.
#### `FreeAbelianQuotient(G): GrpAb -> GrpAb, Map`
The maximal free abelian quotient of the group $G$ as `GrpAb`. The natural epimorphism is returned as second value.
#### `PrimaryAbelianInvariants(A): GrpAb -> [ RngIntElt ]`
#### `AbelianInvariants(A): GrpAb -> [ RngIntElt ]`
The $p$-primary invariants of the abelian group $A$. Each infinite cyclic factor is represented by zero. The non-primary form gives the Smith form invariants, i.e. each element of the sequence divides the next.
#### `PrimaryAbelianBasis(A): GrpAb -> [ GrpAbElt ], [ RngIntElt ]`
#### `AbelianBasis(A): GrpAb -> [ GrpAbElt ], [ RngIntElt ]`
Returns sequences $B$ and $I$, where $I$ are the $p$-primary invariants of $A$, and $B$ are generators for $A$ with orders as in $I$. The non-primary form uses the Smith form invariants, i.e. each element of the sequence divides the next.
#### `TorsionFreeRank(A): GrpAb -> RngIntElt`
The torsion-free rank of the abelian group $G$.
#### `TorsionInvariants(A): GrpAb -> [ RngIntElt ]`
The torsion invariants of the abelian group $G$.
#### `pPrimaryInvariants(A, p): GrpAb, RngIntElt -> [ RngIntElt ]`
The $p$-primary invariants of the abelian group $G$.
### Canonical Decomposition
#### `TorsionFreeSubgroup(A): GrpAb -> GrpAb`
The torsion-free subgroup of the abelian group $G$.
#### `TorsionSubgroup(A): GrpAb -> GrpAb`
The torsion subgroup of the abelian group $G$.
#### `pPrimaryComponent(A, p): GrpAb, RngIntElt -> GrpAb`
The $p$-primary component of the abelian group $G$.
### Set-Theoretic Operations
#### Functions Relating to Group Order
##### `Order(G): GrpAb -> RngIntElt`
##### `# G: GrpAb -> RngIntElt`
The order of the group $G$, returned as an ordinary integer. If $G$ is an infinite group, the value zero is returned. Note that if $G$ is a generic group then determining the order will require the structure of $G$ to be determined.
##### `FactoredOrder(G): GrpAb -> []`
The factored order of the group $G$, returned as a sequence of prime-exponent pairs. If $G$ is an infinite group, the empty sequence is returned. Note that if $G$ is a generic group then determining the order will require the structure of $G$ to be determined.
##### `Exponent(G): GrpAb -> RngIntElt`
The exponent of the group $G$. If the group is infinite, the value zero is returned. Note that if $G$ is a generic group then determining the exponent will require the structure of $G$ to be determined.
##### `IsFinite(G): GrpAb -> BoolElt`
Return `true` if the group $G$ is finite.
##### `IsInfinite(G): GrpAb -> BoolElt`
Return `true` if $G$ is infinite, `false` otherwise.
#### Membership and Equality
##### `g in G: GrpAbElt, GrpAb -> BoolElt`
Given an element $g$ and a group $G$, return `true` if $g$ is an element of $G$, `false` otherwise.
##### `g notin G: GrpAbElt, GrpAb -> BoolElt`
Given an element $g$ and a group $G$, return `true` if $g$ is not an element of $G$, `false` otherwise.
##### `S subset G: { GrpAbElt}, GrpAb -> BoolElt`
Given a group $G$ and a set $S$ of elements belonging to a group $H$, where $G$ and $H$ have some covering group, return `true` if $S$ is a subset of $G$, `false` otherwise.
##### `S notsubset G: { GrpAbElt}, GrpAb -> BoolElt`
Given a group $G$ and a set $S$ of elements belonging to a group $H$, where $G$ and $H$ have some covering group, return `true` if $S$ is not a subset of $G$, `false` otherwise.
##### `H subset G: GrpAb, GrpAb -> BoolElt`
##### `IsSubgroup(H, G): GrpAb, GrpAb -> BoolElt`
Given groups $G$ and $H$, subgroups of some common overgroup, return `true` if $H$ is a subgroup of $G$, and `false` otherwise.
##### `H notsubset G: GrpAb, GrpAb -> BoolElt`
Given groups $G$ and $H$, subgroups of some common overgroup, return `true` if $H$ is not a subgroup of $G$, and `false` otherwise.
##### `G eq H: GrpAb, GrpAb -> BoolElt`
Given groups $G$ and $H$, subgroups of some common overgroup, return `true` if $G$ and $H$ are identical, and `false` otherwise.
##### `G ne H: GrpAb, GrpAb -> BoolElt`
Given groups $G$ and $H$, subgroups of some common overgroup, return `true` if $G$ and $H$ are distinct groups, and `false` otherwise.
#### Set Operations
##### `NumberingMap(G): GrpAb -> Map`
A bijective mapping from the finite group $G$ onto the set of integers $\{ 1 \ldots |G| \rbrace$. The actual mapping depends upon choice of standard generators for $G$.
##### `RandomProcess(G): GrpAb -> Process`
```magma
Slots : RngIntElt Default: 10
Scramble: RngIntElt Default: 100
```
Create a process to generate randomly chosen elements from the finite group $G$. The process is based on the product-replacement algorithm of [[CLGM+95](https://docs.magma-maths.org/references.md#cite-randomelements)], modified by the use of an accumulator. At all times, $N$ elements are stored where $N$ is the maximum of the specified value for `Slots` and `Ngens`$(G) + 1$. Initially, these are just the generators of $G$. As well, one extra group element is stored, the accumulator. Initially, this is the identity. Random elements are now produced by successive calls to `Random(P)`, where $P$ is the process created by this function. Each such call chooses one of the elements in the slots and adds it into the accumulator. The element in that slot is replaced by the sum of it and another randomly chosen slot. The random value returned is the new accumulator value. Setting `Scramble := m` causes $m$ such sum-replacement operations to be performed before the process is returned. Note that this algorithm cannot produce well-distributed random elements of an infinite group.
##### `Random(P): Process -> GrpAbElt`
Given a random element process $P$ created by the function `RandomProcess(G)` for the finite abelian group $G$, return the next random element of $G$ defined by the process.
##### `Random(G): GrpAb -> GrpAbElt`
##### `Random(G, n): GrpAb, RngIntElt -> GrpAbElt`
An element chosen at random from the abelian group $G$. The argument $n$ gives the maximum exponent for infinite generators, and defaults to 10 if not supplied.
##### `Rep(G): GrpAb -> GrpAbElt`
##### `Representative(G): GrpAb -> GrpAbElt`
A representative element of $G$.
### Coset Spaces
#### `Transversal(G, H): GrpAb, GrpAb -> {@ GrpAbElt @}, Map`
#### `RightTransversal(G, H): GrpAb, GrpAb -> {@ GrpAbElt @}, Map`
Given a group $G$ and a subgroup $H$ of $G$, this function returns:
**(a)**
An indexed set of elements $T$ of $G$ forming a right transversal for $G$ over $H$; and,
**(b)**
The corresponding transversal mapping $\phi: G \rightarrow T$. If $T = \lbrace t_1, \ldots, t_r\rbrace$ and $g$ in $G$, $\phi$ is defined by $\phi(g) = t_i$, where $g\in H t_i$.
#### Coercions Between Groups and Subgroups
##### `G ! g: GrpAb, GrpAbElt -> GrpAbElt`
Given an element $g$ belonging to the subgroup $H$ of the group $G$, rewrite $g$ as an element of $G$.
##### `H ! g: GrpAb, GrpAbElt -> GrpAbElt`
Given an element $g$ belonging to the group $G$, and given a subgroup $H$ of $G$ containing $g$, rewrite $g$ as an element of $H$.
##### `K ! g: GrpAb, GrpAbElt -> GrpAbElt`
Given an element $g$ belonging to the group $H$, and a group $K$, such that $H$ and $K$ are subgroups of $G$, and both $H$ and $K$ contain $g$, rewrite $g$ as an element of $K$.
##### `Morphism(H, G): GrpAb, GrpAb -> ModMatRngElt`
The integer matrix defining the inclusion monomorphism from the subgroup $H$ of $G$ into $G$.
### Subgroup Constructions
Although, in the case of an abelian group, many of the standard subgroup constructors are trivial, they are all implemented for the sake of uniformity. Here we document only those which are meaningful in the context of abelian groups.
#### `H meet K: GrpAb, GrpAb -> GrpAb`
Given subgroups $H$ and $K$ of some group $G$, construct their intersection.
#### `H meet:= K: GrpAb, GrpAb -> GrpAb`
Replace $H$ with the intersection of groups $H$ and $K$.
#### `H + K: GrpAb, GrpAb -> GrpAb`
Given subgroups $H$ and $K$ of some group $G$, construct the smallest subgroup containing both.
#### `n * G: RngIntElt, GrpAb -> GrpAb, Map`
For an integer $n$ and some abelian group $G$, construct the subgroup $nG$. The second return value is the map $G \to G$ sending $g \to ng$.
#### `FrattiniSubgroup(G): GrpAb -> GrpAb`
The Frattini subgroup of the finite abelian group $G$.
#### `SylowSubgroup(G, p : parameters): GrpAb, RngIntElt -> GrpAb`
#### `Sylow(G, p : parameters): GrpAb, RngIntElt -> GrpAb`
```magma
Structure: Bool Default: false
```
The Sylow $p$-subgroup for the group $G$. If $G$ is a generic group and the parameter `Structure` is true, or if the group structure of $A$ is known, then the group structure of the Sylow subgroup is computed.
#### `Example: Sylow 2-Subgroup of a Unit Group (ex-c74c85)`
In the following example, we construct the Sylow $2$-subgroup of $G = Z_{34384}$.
```magma
> m := 34384;
> Zm := Integers(m);
> U := {@ x : x in Zm | GCD(x, m) eq 1 @};
> G := GenericAbelianGroup(U : IdIntrinsic := "Id",
> AddIntrinsic := "*", InverseIntrinsic := "/");
> _ := AbelianGroup(G);
> Factorization(#G);
> Sylow(G, 2);
2-Sylow subgroup: Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/2 + Z/2 + Z/2 + Z/4
Defined on 4 generators in supergroup G:
GAp.1 = G.1
GAp.2 = G.2
GAp.3 = 3*G.3
GAp.4 = 153*G.4
Relations:
2*GAp.1 = 0
2*GAp.2 = 0
2*GAp.3 = 0
4*GAp.4 = 0
```
### Subgroup Chains
Series involving centres or commutator subgroups are likely to be trivial for abelian groups, but are noted here for completeness.
#### `ChiefSeries(G): GrpAb -> [GrpAb]`
#### `CompositionSeries(G): GrpAb -> [GrpAb]`
A composition series for the finite abelian group $G$ returned as a sequence of subgroups.
#### `Agemo(G, i): GrpAb, RngIntElt -> GrpAb`
Given a finite $p$-group $G$, return the characteristic subgroup of $G$ generated by the elements $x^{p^i}$, $x \in G$, where $i$ is a positive integer.
#### `Omega(G, i): GrpAb, RngIntElt -> GrpAb`
Given a finite $p$-group $G$, return the characteristic subgroup of $G$ generated by the elements of order dividing $p^i$, where $i$ is a positive integer.
#### `ElementaryAbelianSeries(G): GrpAb -> [ GrpAb ]`
A descending series of subgroups of $G$ where each quotient is elementary abelian.
#### `DerivedSeries(G): GrpAb -> [ GrpAb ]`
#### `LowerCentralSeries(G): GrpAb -> [ GrpAb ]`
The derived series of $G$.
#### `UpperCentralSeries(G): GrpAb -> [ GrpAb ]`
The upper central series of $G$.
#### `SubnormalSeries(G, H): GrpAb, GrpAb -> [ GrpAb ]`
A subnormal series from $G$ to the subnormal subgroup $H$.
### General Group Properties
#### `IsCyclic(G): GrpAb -> BoolElt`
Returns `true` if the group $G$ is cyclic, `false` otherwise.
#### `IsElementaryAbelian(G): GrpAb -> BoolElt`
Returns `true` if the group $G$ is elementary abelian, `false` otherwise.
#### `IsFree(G): GrpAb -> BoolElt`
Returns `true` if $G$ is free, `false` otherwise.
#### `IsMixed(G): GrpAb -> BoolElt`
Returns `true` if $G$ is a mixed group, `false` otherwise. An abelian group is mixed if it is neither a torsion group nor free.
#### `IspGroup(G): GrpAb -> BoolElt`
Returns `true` if the finite group $G$ is a $p$-group, i.e. if all elements have order a power of $p$.
#### `DerivedLength(G): GrpAb -> RngIntElt`
The length of the derived series of $G$.
#### Properties of Subgroups
##### `IsMaximal(G, H): GrpAb, GrpAb -> BoolElt`
Returns `true` if the subgroup $H$ of the finite group $G$ is a maximal subgroup of $G$, `false` otherwise.
##### `Index(G, H): GrpAb, GrpAb -> RngIntElt`
The index of the subgroup $H$ in the group $G$, returned as an ordinary integer. If $H$ has infinite index in $G$, the value zero is returned.
##### `FactoredIndex(G, H): GrpAb, GrpAb -> []`
The factored index of the subgroup $H$ in the group $G$, returned as a sequence of prime-exponent pairs. If $H$ has infinite index in $G$, the empty sequence is returned.
##### `IsPure(G, H): GrpAb, GrpAb -> BoolElt`
Returns `true` if the subgroup $H$ of the finite group $G$ is pure, ie. if for all $n$ we have $nG \cap H = nH$.
##### `IsNeat(G, H): GrpAb, GrpAb -> BoolElt`
Returns `true` if the subgroup $H$ of the finite group $G$ is neat, i.e., if for all primes $p$ we have $pG \cap H = pH$.
#### Enumeration of Subgroups
##### `MaximalSubgroups(G): GrpAb -> [GrpAb]`
The maximal subgroups of the finite group $G$ returned as a sequence of subgroups.
##### `Subgroups(G:parameters): GrpAb -> [Rec]`
```magma
Sub : [RngIntElt] Default: []
Quot: [RngIntElt] Default: []
```
The subgroups of the finite group $G$ are returned as a sequence of records. The record fields are `subgroup`, storing the actual group; `order`, storing the group order; and `length`, storing the length of the conjugacy class, which is always 1 for abelian groups.
If the parameter `Sub` is set, only subgroups with invariants equal to the given sequence are found. The given sequence should contain positive integers, such that each divides the following.
If the parameter `Quot` is set, only subgroups such that the quotient group has invariants equal to the given sequence are found. The given sequence should contain positive integers, such that each divides the following.
##### `NumberOfSubgroupsAbelianPGroup(A): SeqEnum -> SeqEnum`
Return the number of subgroups of each non-trivial order in the abelian $p$-group $G$ where $A = [a_1, a_2, \ldots]$ and $G = C_{a_1} \times C_{a_2} \times \ldots$. The $m$-th entry in the sequence returned is the number of subgroups of order $p^m$.
##### `HasComplement(G, U): GrpAb, GrpAb -> BoolElt, GrpAb`
For a finite abelian group $G$ and a subgroup $U$ decide if there exist some other subgroup $V$ such that $G = U+V$ and $U \cap V = \lbrace 0\rbrace$. In case such a $V$ exists, it is returned as the second value.
##### `Example: Subgroups of an Abelian Group of Order 12 (ex-604d09)`
We look at subgroups of an abelian group of order 12.
```magma
> G := AbelianGroup([2,6]);
> s := Subgroups(G); #s;
10
> s[7];
rec |
order := 3, length := 1,
subgroup := Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup G:
$.1 = 2*G.2
Relations:
3*$.1 = 0>
> [x`order:x in s];
[ 12, 6, 4, 2, 6, 6, 3, 2, 2, 1 ]
```
Now we find the elementary abelian subgroup of order 4.
```magma
> s22 := Subgroups(G:Sub := [2,2]); #s22;
1
> s22;
Conjugacy classes of subgroups
------------------------------
[1] Order 4 Length 1
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators in supergroup G:
$.1 = G.1
$.2 = 3*G.2
Relations:
2*$.1 = 0
2*$.2 = 0
```
There is more than one subgroup of index 2 in $G$.
```magma
> q2 := Subgroups(G:Quot := [2]); #q2;
3
> q2[3]`subgroup;
Abelian Group isomorphic to Z/6
Defined on 1 generator in supergroup G:
$.1 = G.1 + G.2
Relations:
6*$.1 = 0
```
### Representation Theory
#### `CharacterTable(G): GrpAb -> TabChtr`
The table of irreducible characters for the group $G$.
### The Hom Functor
#### `Hom(G, H): GrpPC, GrpPC -> GrpAb, Map`
#### `Hom(G, H): GrpAb, GrpAb -> GrpAb, Map`
Given finite abelian groups $G$ and $H$, return an abelian group $A$ isomorphic to ${\rm Hom}(G, H)$, and a transfer map $t$ such that, given an element $a$ of $A$, $t(a)$ yields the corresponding (Magma `Map` type) homomorphism from $G$ to $H$. The structure of ${\rm Hom}(G, H)$ may thus be analyzed by examining $A$.
#### `HomGenerators(G, H): GrpAb, GrpAb -> GrpAb, Map`
Given finite abelian groups $G$ and $H$, return a sequence of ($Z$-module) generators of ${\rm Hom}(G, H)$. The generators are returned as actual (Magma `Map` type) homomorphisms. Note that ${\rm Hom}(G, H)$ is usually not free, so it is difficult to generate all homomorphisms uniquely using the generators alone (use `Hom` or `Homomorphisms` if that is desired).
#### `AllHomomorphisms(G, H): GrpAb, GrpAb -> [Map]`
#### `AllHomomorphisms(G, H): GrpPC, GrpPC -> [Map]`
#### `Homomorphisms(G, H): GrpAb, GrpAb -> [Map]`
Given finite abelian groups $G$ and $H$, return a sequence containing all elements of ${\rm Hom}(G, H)$. The elements are returned as actual (Magma `Map` type) homomorphisms. Note that this function simply uses `Hom`, transferring each element of the returned group to the actual Magma `Map` type homomorphism.
#### `Example: Hom Functor on Abelian Groups (ex-0d5388)`
We examine $A={\rm Hom}(G, H)$, for certain abelian groups $G$ and $H$.
```magma
> G := AbelianGroup([2, 3]);
> H := AbelianGroup([4, 6]);
> A, t := Hom(G, H);
> #A;
12
> A;
Abelian Group isomorphic to Z/2 + Z/6
Defined on 2 generators
Relations:
2*A.1 = 0
6*A.2 = 0
> h := t(A.1);
> h;
Mapping from: GrpAb: G to GrpAb: H
> h(G.1);
3*H.2
> h(G.2);
0
```
We now enumerate all elements of $A$ and examine the images of each generator of $G$ under each homomorphism. We note that each possible list of images occurs only once.
```magma
> I := [ where h is t(x): x in A];
> I;
[
<0, 0>,
<3*H.2, 0>,
<2*H.1, 2*H.2>,
<2*H.1 + 3*H.2, 2*H.2>,
<0, 4*H.2>,
<3*H.2, 4*H.2>,
<2*H.1, 0>,
<2*H.1 + 3*H.2, 0>,
<0, 2*H.2>,
<3*H.2, 2*H.2>,
<2*H.1, 4*H.2>,
<2*H.1 + 3*H.2, 4*H.2>
]
> #I;
12
> #Set(I);
12
```
### Automorphism Groups
#### `AutomorphismGroup(G): GrpAb -> GrpAuto`
The full automorphism group of the abelian group $G$.
#### `IsIsomorphic(G, H): GrpAb, GrpAb -> BoolElt, Hom(Grp)`
Test whether or not the two abelian groups $G$ and $H$ are isomorphic as abstract groups. If so, both the result `true` and an isomorphism from $G$ to $H$ is returned. If not, the result `false` is returned.
### Cohomology
#### `Dual(G): GrpAb -> GrpAb, Map`
Computes the dual group $G^*$ of $G$ and a map $M$ from $G\times G^* \to {\mathbb{Z}}/m{\mathbb{Z}}$ for $m$ the exponent of $G$ that allows $G^*$ to act on $G$. The group $G$ must be finite.
#### `H2_G_QmodZ(G): GrpAb -> GrpAb, Map`
Computes $H := H^2(G, {\mathbb{Q}}/{\mathbb{Z}})$ and a map $f : H \to (G\times G \to {\mathbb{Z}}/m{\mathbb{Z}})$ that will give the cocycles as maps from $G\times G \to {\mathbb{Z}}/m{\mathbb{Z}}$, $m := \#G$.
#### `Res_H2_G_QmodZ(U, H2): GrpAb, GrpAb -> GrpAb, Map`
For a subgroup $U$ of $G$ and $H2 = H^2(G, {\mathbb{Q}}/{\mathbb{Z}})$ computes $H^2(U, {\mathbb{Q}}/{\mathbb{Z}})$ in a compatible way together with the restriction map into $H2$.
The abelian group $H2$ must be the result of `H2_G_QmodZ` as this function relies on the attributes stored in there.
### Homomorphisms
Two functions are provided to construct homomorphisms or isomorphisms from one group into another, where either of the groups, or both, may be generic abelian groups.
#### `hom< A -> B | L>: Grp, Grp, List -> Map`
Given groups $A$ and $B$, construct a homomorphism from $A$ to $B$ as defined by the extension $L$. If one or both of $A$ and $B$ are generic abelian groups this works as usual, with one minor difference as explained below. Suppose that the generators of $A$ are $g_1, \dots, g_n$, and that $\phi(g_i)=h_i$ for each $i$, where $\phi$ is the homomorphism one wishes to construct. The list $L$ as required by the constructors must be one of the following:
**(a)**
a list of the $n$ 2-tuples $< g_i, h_i >$ (order not important);
**(b)**
a list of the $n$ arrow-pairs $g_i \rightarrow h_i$ (order not important);
**(c)**
$h_1, \dots, h_n$ (order is important).
If $A$ is a generic abelian group this rule is relaxed somewhat in the following sense: If $L$ is a list of $n$ 2-tuples or of $n$ arrow-pairs, the elements $g_i$ need not be the defining generators of $A$. The only requirement is that the set $\{g_1, \ldots, g_n\}$ does actually generate the whole of $A$.
#### `Homomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] -> Map`
Creates a homomorphism from $A$ into $B$ as given by the mapping of $X$ into $Y$. The arguments $A$ and $B$ may be any type of group, including of course, generic abelian groups.
The function `Homomorphism` does not require the elements of the argument $X$ to be generators of $A$ as given by `Generators(A)`, so it allows more freedom when creating a homomorphism. If, however, these elements fail to generate the whole of $A$ then the subsequent map application will fail.
#### `iso< A -> B | L>: Grp, Grp, List -> Map`
Given groups $A$ and $B$, construct an isomorphism from $A$ to $B$ as defined by the extension $L$. If one or both of $A$ and $B$ are generic abelian groups this works as usual, with one minor difference as explained below. Suppose that the generators of $A$ are $g_1, \dots, g_n$, and that $\phi(g_i)=h_i$ for each $i$, where $\phi$ is the isomorphism one wishes to construct. The list $L$ as required by the constructors must be one of the following:
**(a)**
a list of the $n$ 2-tuples $< g_i, h_i >$ (order not important);
**(b)**
a list of the $n$ arrow-pairs $g_i \rightarrow h_i$ (order not important);
**(c)**
$h_1, \dots, h_n$ (order is important).
If $A$ is a generic abelian group this rule is relaxed somewhat in the following sense: If $L$ is a list of $n$ 2-tuples or of $n$ arrow-pairs, the elements $g_i$ may not necessarily be generators of $A$ as given by the function `Generators(A)`. The only requirement is that the set $\{h_1, \ldots, h_n\}$ does actually generate the whole of $B$.
#### `Isomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] -> Map`
Creates a isomorphism from $A$ into $B$ as given by the mapping of $X$ into $Y$. The arguments $A$ and $B$ can be any type of group, including of course, generic abelian groups.
The function `Isomorphism` does not require the elements of the argument $X$ to be generators of $A$ as given by `Generators(A)`, so it allows more freedom when creating an isomorphism. If, however, these elements fail to generate the whole of $A$ then the subsequent map application will fail.
#### `Example: Homomorphisms of Generic Abelian Groups (ex-32bea9)`
Recall that we defined the subgroups $GH1_{Zm}$ and $GH2_{Zm}$ of $G$ as:
```magma
> GH1_Zm;
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/6 + Z/612
Defined on 2 generators in supergroup G:
GH1_Zm.1 = G.2 + G.3
GH1_Zm.2 = G.4
Relations:
6*GH1_Zm.1 = 0
612*GH1_Zm.2 = 0
> GH2_Zm;
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/2 + Z/2 + Z/6 + Z/612
Defined on 4 generators in supergroup G:
GH2_Zm.1 = G.1
GH2_Zm.2 = G.2
GH2_Zm.3 = G.3
GH2_Zm.4 = G
Relations:
2*GH2_Zm.1 = 0
2*GH2_Zm.2 = 0
6*GH2_Zm.3 = 0
612*GH2_Zm.4 = 0
```
We construct the homomorphism
```magma
> h := hom GH2_Zm | GH2_Zm.1, GH2_Zm.2 >;
> h(GH1_Zm);
Generic Abelian Group over
Residue class ring of integers modulo 34384
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators in supergroup GH2_Zm:
$.1 = GH2_Zm.2
$.2 = GH2_Zm.1
Relations:
2*$.1 = 0
2*$.2 = 0
```
but we cannot construct the isomorphism
```magma
> i := iso GH2_Zm | GH2_Zm.1, GH2_Zm.2 >;
>> i := iso GH2_Zm | GH2_Zm.1, GH2_Zm.2 >;
^
Runtime error in map< ... >: Images do not generate the (whole) codomain
```
An alternative way of creating the homomorphism $h$ would be
```magma
> h := Homomorphism(GH1_Zm, GH2_Zm, Generators(GH1_Zm), [GH2_Zm.1, GH2_Zm.2]);
```
## Free Groups
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/introduction.md)
- [Free Groups and Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md)
- [Construction of a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#construction-of-a-free-group)
- [`FreeGroup(n): RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-grpfp-1-freegroup)
- [Example: Creating a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-f04611)
- [Construction of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#construction-of-words)
- [`G ! [ i₁, ..., iₛ ]: GrpFP, [ RngIntElt ] → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-grpfp-rngintelt)
- [`Identity(G): GrpFP → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-identity-grpfp)
- [`Id(G): GrpFP → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-id-grpfp)
- [`G ! 1: GrpFP, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-grpfp-rngintelt-2)
- [`Random(G, m, n): GrpFP, RngIntElt, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-random-grpfp-rngintelt-rngintelt)
- [Access Functions for Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#access-functions-for-words)
- [`# w: GrpFPElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-operation-grpfpelt-rngintelt)
- [`ElementToSequence(w): GrpFPElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-elementtosequence-grpfpelt)
- [`Eltseq(w): GrpFPElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-eltseq-grpfpelt)
- [`ExponentSum(w, x): GrpFPElt, GrpFPElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-exponentsum-grpfpelt-grpfpelt)
- [`Weight(w, x): GrpFPElt, GrpFPElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-weight-grpfpelt-grpfpelt)
- [`GeneratorNumber(w): GrpFPElt → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-generatornumber-grpfpelt)
- [`LeadingGenerator(w): GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-grpfp-1-leadinggenerator)
- [`Parent(w): GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-parent-grpfpelt)
- [Example: Abelianised Relation Matrix from Word Weights](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-117fb7)
- [Arithmetic Operators for Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#arithmetic-operators-for-words)
- [`u * v: GrpFPElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-times-grpfpelt-grpfpelt)
- [`u ^ n: GrpFPElt, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-pow-grpfpelt-rngintelt)
- [`u ^ v: GrpFPElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-pow-grpfpelt-grpfpelt)
- [`(u, v): GrpFPElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#literal-literal-u-v-grpfpelt-grpfpelt-grpfpelt)
- [`(u₁, ..., uₙ): List(GrpFPElt) → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#literal-literal-u1-un-list-grpfpelt-grpfpelt)
- [Comparison of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#comparison-of-words)
- [`u eq v: GrpFPElt, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-eq-grpfpelt-grpfpelt)
- [`u ne v: GrpFPElt, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-ne-grpfpelt-grpfpelt)
- [`u lt v: GrpFPElt, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-lt-grpfpelt-grpfpelt)
- [`u le v: GrpFPElt, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-le-grpfpelt-grpfpelt)
- [`u ge v: GrpFPElt, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-ge-grpfpelt-grpfpelt)
- [`u gt v: GrpFPElt, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#operation-op-gt-grpfpelt-grpfpelt)
- [Example: Length, Weight, and Leading Generator of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-65f8bd)
- [String Operations on Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#string-operations-on-words)
- [`Eliminate(u, x, v): GrpFPElt, GrpFPElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-eliminate-grpfpelt-grpfpelt-grpfpelt)
- [`Eliminate(U, x, v): { GrpFPElt }, GrpFPElt, GrpFPElt → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-eliminate-grpfpelt-grpfpelt-grpfpelt-2)
- [`Match(u, v, f): GrpFPElt, GrpFPElt, RngIntElt → BoolElt, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-match-grpfpelt-grpfpelt-rngintelt)
- [`RotateWord(u, n): GrpFPElt, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-rotateword-grpfpelt-rngintelt)
- [`Substitute(u, f, n, v): GrpFPElt, RngIntElt, RngIntElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-substitute-grpfpelt-rngintelt-rngintelt-grpfpelt)
- [`Subword(u, f, n): GrpFPElt, RngIntElt, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-subword-grpfpelt-rngintelt-rngintelt)
- [Example: String Operations on Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-b14d33)
- [Finitely Generated Subgroups of Free Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md)
- [`Supergroup(F): GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-supergroup-grpfp)
- [`x in H: GrpFP, GrpFPElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#operation-op-in-grpfp-grpfpelt)
- [`IsSubgroup(H, K): GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-issubgroup-grpfp-grpfp)
- [`H eq K: GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#operation-op-eq-grpfp-grpfp)
- [`Index(F, H): GrpFP, GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-index-grpfp-grpfp)
- [`HasFiniteIndex(F, H): GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-hasfiniteindex-grpfp-grpfp)
- [`FreeGenerators(H): GrpFP → SeqEnum, GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-freegenerators-grpfp)
- [`Rewrite(H, x): GrpFP, GrpFPElt → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-rewrite-grpfp-grpfpelt)
- [`SubgroupElements(G, H, n): GrpFP, GrpFP, RngIntElt → SetEnum`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-subgroupelements-grpfp-grpfp-rngintelt)
- [`H meet K: GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#operation-op-meet-grpfp-grpfp)
- [`Centraliser(F,x): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-centraliser-grpfp-grpfpelt)
- [`Centralizer(F,x): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-centralizer-grpfp-grpfpelt)
- [`IsConjugate(F, x, y): GrpFP, GrpFPElt, GrpFPElt → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-isconjugate-grpfp-grpfpelt-grpfpelt)
- [`Centraliser(F, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-centraliser-grpfp-grpfp)
- [`Centralizer(F, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-centralizer-grpfp-grpfp)
- [`Normaliser(F, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-normaliser-grpfp-grpfp)
- [`Normalizer(F, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-normalizer-grpfp-grpfp)
- [`IsConjugate(F, H, K): GrpFP, GrpFP, GrpFP → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#function-isconjugate-grpfp-grpfp-grpfp)
- [Example: Subgroups of a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/subfree.md#example-ex-f8ece0)
- [The Automorphism Group of a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/autfree.md)
- [`AutomorphismGroup(F): GrpFP → GrpAuto`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/autfree.md#function-automorphismgroup-grpfp)
- [`InverseAutomorphismFreeGroup(F, Q): GrpFP, SeqEnum → GrpAutoElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/autfree.md#function-inverseautomorphismfreegroup-grpfp-seqenum)
- [`WhiteheadReduction(F, Q): GrpFP, SeqEnum → BoolElt, SeqEnum, GrpAutoElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/autfree.md#function-whiteheadreduction-grpfp-seqenum)
- [Example: Automorphism Group of a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/autfree.md#example-ex-baab12)
- [Example: Automorphism Group of a Free Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/autfree.md#example-ex-1a5108)
- [Bibliography](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/bibliography.md)
### Introduction
This chapter presents the functions designed for computing with free groups and their words.
### Free Groups and Words
#### Construction of a Free Group
##### `FreeGroup(n): RngIntElt -> GrpFP`
Construct the free group $F$ of rank $n$, where $n$ is a positive integer.
The $i$-th generator of $F$ may be referenced by the expression `F.i`, $i = 1, \ldots, n$. Note that a special form of the assignment statement is provided which enables the user to assign names to the generators of $F$. In this form of assignment, the list of generator names is enclosed within angle brackets and appended to the variable name on the *left hand side* of the assignment statement:
`F< v_1, ... v_n > := FreeGroup(n);`
##### `Example: Creating a Free Group (ex-f04611)`
The statement
```magma
> F := FreeGroup(2);
```
creates the free group of rank 2. Here the generators may be referenced using the standard names, `F.1` and `F.2`. The statement
```magma
> F := FreeGroup(2);
```
defines $F$ to be the free group of rank 2 and assigns the names $x$ and $y$ to the two generators.
#### Construction of Words
The operations in this section apply to both, free groups and arbitrary fp-groups.
##### `G ! [ i₁, ..., iₛ ]: GrpFP, [ RngIntElt ] -> GrpFPElt`
Given a group $G$ defined on $r$ generators and a sequence $[i_1, \cdots, i_s]$ of integers lying in the range $[-r, r]$, excluding $0$, construct the word
$$
G.|i_1|^{\epsilon_1}*G.|i_2|^{\epsilon_2}* \cdots *G.|i_s|^{\epsilon_s}
$$
where $\epsilon_j$ is $+1$ if $i_j$ is positive, and $-1$ if $i_j$ is negative.
##### `Identity(G): GrpFP -> GrpFPElt`
##### `Id(G): GrpFP -> GrpFPElt`
##### `G ! 1: GrpFP, RngIntElt -> GrpFPElt`
Construct the identity element, represented as the empty word, for the fp-group $G$. For a sample application of this function, see Example [Length, Weight, and Leading Generator of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-65f8bd).
##### `Random(G, m, n): GrpFP, RngIntElt, RngIntElt -> GrpFPElt`
A random word of length $l$ in the generators of the group $G$, where $m \leq l \leq n$. For a sample application of this function, see Example [Length, Weight, and Leading Generator of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-65f8bd).
#### Access Functions for Words
This section describes some basic access functions for words. These operations apply to both, free groups and arbitrary fp-groups.
##### `# w: GrpFPElt -> RngIntElt`
The length of the word $w$.
##### `ElementToSequence(w): GrpFPElt -> [ RngIntElt ]`
##### `Eltseq(w): GrpFPElt -> [ RngIntElt ]`
The sequence $Q$ obtained by decomposing the word $w$ into its constituent generators and generator inverses. Suppose $w$ is a word in the group $G$. Then, if $w = G.i_1^{e_1} \cdots G.i_m^{e_m}$, with each $e_i = \pm 1$, then $Q[j] = i_j$ if $e_j = +1$ and $Q[j] = -i_j$ if $e_j = {-1}$, for $j = 1, \ldots, m$.
##### `ExponentSum(w, x): GrpFPElt, GrpFPElt -> RngIntElt`
##### `Weight(w, x): GrpFPElt, GrpFPElt -> RngIntElt`
Given a word $w$, and the name of a generator $x$ of a group $G$, compute the sum of the exponents of the generator $x$ in the word $w$. For a sample application of this function, see Example [Length, Weight, and Leading Generator of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-65f8bd).
##### `GeneratorNumber(w): GrpFPElt -> RngIntElt`
Suppose $w$ is a word belonging to a group $G$. Assume $x$ is the name of the $i$-th generator of $G$. Then
**(i)**
if $w =$ `Identity(G)`, `GeneratorNumber(w)` is 0;
**(ii)**
if $w = x*w'$, $w'$ a word in $G$, `GeneratorNumber(w)` is $i$;
**(iii)**
if $w = x^{-1}*w'$, $w'$ a word in $G$, `GeneratorNumber(w)` is $-i$.
##### `LeadingGenerator(w): GrpFPElt -> GrpFPElt`
Suppose $w$ is a word belonging to a group $G$. If $w = x^\epsilon*w'$, $w'$ a word in $G$, $x$ a generator of $G$, and $\epsilon\in\{-1,+1\}$, the functions returns $x^\epsilon$. If $w=$ `Identity(G)`, it returns `Identity(G)`. For a sample application of this function, see Example [Length, Weight, and Leading Generator of Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#example-ex-65f8bd).
##### `Parent(w): GrpFPElt -> GrpFP`
The parent group $G$ of the word $w$.
##### `Example: Abelianised Relation Matrix from Word Weights (ex-117fb7)`
Consider the free group $F$ on 6 generators
```magma
> F := FreeGroup(6);
```
and the sequence of words
```magma
> rels := [ (u*v)^42, (v,x), (x*z^2)^4,
> v^2*y^3, (v*z)^3, y^4, (x*z)^3 ];
```
The abelianised relation matrix of the quotient
$$
\langle u,v,w,x,y,z \;\;|\;\;
(uv)^{42}, (v,x), (xz^2)^4, v^2y^3, (vz)^3, y^4, (xz)^3 \rangle
$$
can be obtained using the following construction
```magma
> R := Matrix(Integers(),
> [ [ Weight(r, F.j) : j in [1..6] ] : r in rels ]);
> R;
[42 42 0 0 0 0]
[ 0 0 0 0 0 0]
[ 0 0 0 4 0 8]
[ 0 2 0 0 3 0]
[ 0 3 0 0 0 3]
[ 0 0 0 0 4 0]
[ 0 0 0 3 0 3]
```
(The function `Matrix` constructs a matrix from a sequence of row vectors.)
#### Arithmetic Operators for Words
Suppose $G$ is an fp-group for which generators have already been defined. This subsection defines the elementary arithmetic operations on words that are derived from the multiplication and inversion operators. The availability of the operators defined here enables the user to construct an element (*word*) of $G$ in terms of the generators as follows:
**(i)**
A generator is a word;
**(ii)**
The expression $(u)$ is a word, where $u$ is a word;
**(iii)**
The product $u*v$ of the words $u$ and $v$ is a word;
**(iv)**
The conjugate $u^v$ of the word $u$ by the word $v$, is a word ($u^v$ expands into the word $v^{-1}*u*v$);
**(v)**
The power of a word, $u^n$, where $u$ is a word and $n$ is an integer, is a word;
**(vi)**
The commutator $(u, v)$ of the words $u$ and $v$ is a word ($(u, v)$ expands into the word $u^{-1}*v^{-1}*u*v$). Note that $(u, v, w)$ is equivalent to $((u, v), w)$, i.e. commutators are *left-normed*.
The word operations defined here may be applied either to the words of a free group or the words of a group with non-trivial relations. If such an operator is applied to a group possessing non-trivial relations, only free reduction will be applied to the resulting words.
##### `u * v: GrpFPElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$ belonging to the same fp-group $G$, return the product of $u$ and $v$.
##### `u ^ n: GrpFPElt, RngIntElt -> GrpFPElt`
The $n$-th power of the word $u$, where $n$ is an integer. When invoked with $n = {-1}$, the function computes the inverse of $u$. When invoked with $n = 0$, the function returns the identity element.
##### `u ^ v: GrpFPElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$ belonging to the same fp-group $G$, return the conjugate $v^{-1}*u*v$ of the word $u$ by the word $v$.
##### `(u, v): GrpFPElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$ belonging to the same fp-group $G$, return the commutator $u^{-1}v^{-1}uv$ of the words $u$ and $v$.
##### `(u₁, ..., uₙ): List(GrpFPElt) -> GrpFPElt`
Given the $n$ words $u_1, \ldots, u_n$ belonging to the same fp-group $G$, return their commutator. Commutators are *left-normed*, so that they are evaluated from left to right.
#### Comparison of Words
Words in an fp-group may be compared both for equality and for their relationship with respect to a natural lexicographic ordering. It should be noted that even when a pair of words belong to a group defined by non-trivial relations, only the free reductions of the words are compared. Thus, a pair of words belonging to a group $G$ may be declared to be distinct even though they may represent the same element of $G$. The words of an fp-group $G$ are ordered first by length and then lexicographically. The lexicographic ordering is determined by the following ordering on the generators and their inverses:
$$
G.1 < G.1^{-1} < G.2 < G.2^{-1} < \cdots
$$
Here, $u$ and $v$ are words belonging to some common fp-group.
##### `u eq v: GrpFPElt, GrpFPElt -> BoolElt`
Return `true` if the free reductions of the words $u$ and $v$ are identical.
##### `u ne v: GrpFPElt, GrpFPElt -> BoolElt`
Return `true` if the free reductions of the words $u$ and $v$ are not identical.
##### `u lt v: GrpFPElt, GrpFPElt -> BoolElt`
Return `true` if the word $u$ precedes the word $v$, with respect to the ordering defined above for elements of an fp-group. The words $u$ and $v$ are freely reduced before the comparison is made.
##### `u le v: GrpFPElt, GrpFPElt -> BoolElt`
Return `true` if the word $u$ either precedes, or is equal to, the word $v$, with respect to the ordering defined above for elements of an fp-group. The words $u$ and $v$ are freely reduced before the comparison is made.
##### `u ge v: GrpFPElt, GrpFPElt -> BoolElt`
Return `true` if the word $u$ either follows, or is equal to, the word $v$, with respect to the ordering defined above for elements of an fp-group. The words $u$ and $v$ are freely reduced before the comparison is made.
##### `u gt v: GrpFPElt, GrpFPElt -> BoolElt`
Return `true` if the word $u$ follows the word $v$, with respect to the ordering defined above for elements of an fp-group. The words $u$ and $v$ are freely reduced before the comparison is made.
##### `Example: Length, Weight, and Leading Generator of Words (ex-65f8bd)`
We construct the free group on three generators and generate a random word $w$ of length between 4 and 6.
```magma
> F := FreeGroup(3);
>
> w := Random(F, 4, 6);
> w;
b^-1 * a^-1 * b^2 * a^-1
```
We print the length of $w$ and the weight (the exponent sum) of generator $a$ in $w$.
```magma
> #w;
5
>
> Weight(w, a);
-2
```
We now strip the generators from $w$ one by one, using arithmetic and comparison operators for words and the access function [`LeadingGenerator`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-grpfp-1-leadinggenerator).
```magma
> while w ne Identity(F) do
> g := LeadingGenerator(w);
> print g;
> w := g^-1 * w;
> end while;
b^-1
a^-1
b
b
a^-1
```
#### String Operations on Words
The functions described in this section perform low level string operations like substitution, elimination, or substring matching on elements of fp-groups.
##### `Eliminate(u, x, v): GrpFPElt, GrpFPElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$, and a generator $x$, all belonging to a group $G$, return the word obtained from $u$ by replacing each occurrence of $x$ by $v$ and each occurrence of $x^{-1}$ by $v^{-1}$.
##### `Eliminate(U, x, v): { GrpFPElt }, GrpFPElt, GrpFPElt -> { GrpFPElt }`
Given a set of words $U$, a word $v$, and a generator $x$, all belonging to a group $G$, return the set of words obtained by taking each element $u$ of $U$ in turn, and replacing each occurrence of $x$ in $u$ by $v$ and each occurrence of $x^{-1}$ by $v^{-1}$.
##### `Match(u, v, f): GrpFPElt, GrpFPElt, RngIntElt -> BoolElt, RngIntElt`
Suppose $u$ and $v$ are words belonging to the same group $G$, and that $f$ is an integer such that $1 \leq f \leq \# u$. The function seeks the least integer $l$ such that:
**(a)**
$l\geq f$; and
**(b)**
$v$ appears as a subword of $u$, starting at the $l$-th letter of $u$.
If such an integer $l$ is found `Match` returns the value `true` and $l$. If no such $l$ is found, `Match` returns the value `false`.
##### `RotateWord(u, n): GrpFPElt, RngIntElt -> GrpFPElt`
The word obtained by cyclically permuting the word $u$ by $n$ places. If $n$ is positive, the rotation is from left to right, while if $n$ is negative the rotation is from right to left. In the case where $n$ is zero, the function returns $u$.
##### `Substitute(u, f, n, v): GrpFPElt, RngIntElt, RngIntElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$ belonging to a group $G$, and non-negative integers $f$ and $n$, this function replaces the substring of $u$ of length $n$, starting at position $f$, by the word $v$. Thus, if $u = x_{i_1}^{e_1}\cdots x_{i_f}^{e_f}\cdots x_{i_{f+n-1}}^{e_{f+n-1}}\cdots x_{i_m}^{e_m}$ then the substring $x_{i_f}^{e_f}\cdots x_{i_{f+n-1}}^{e_{f+n-1}}$ is replaced by $v$. If the function is invoked with $v =$ Id(G), then the substring $x_{i_f}^{e_f}\cdots x_{i_{f+n-1}}^{e_{f+n-1}}$ of $u$ is deleted.
##### `Subword(u, f, n): GrpFPElt, RngIntElt, RngIntElt -> GrpFPElt`
The subword of the word $u$ comprising the $n$ consecutive letters commencing at the $f$-th letter of $u$.
##### `Example: String Operations on Words (ex-b14d33)`
We demonstrate some of these operations in the context of the free group on generators $x$, $y$, and $z$.
```magma
> F := FreeGroup(3);
> u := (x, y*z);
> w := u^(x^2*y);
> #w;
12
> w;
y^-1 * x^-3 * z^-1 * y^-1 * x * y * z * x^2 * y
```
We replace each occurrence of the generator $x$ in $w$ by the word $y*z^-1$.
```magma
> Eliminate(w, x, y*z^-1);
y^-1 * z * y^-1 * z * y^-1 * z * y^-1 * z^-2 * y * z * y * z^-1 * y * z^-1 * y
```
We count the number of occurrences of each generator in $w$.
```magma
> [ ExponentSum(w, F.i) : i in [1..Ngens(F)] ];
[ 0, 0, 0 ]
> GeneratorNumber(w);
-2
```
We locate the start of the word $u$ in the word $w$.
```magma
> b, p := Match(w, u, 1);
> b, p;
true 4
```
We now replace the subword $u$ in $w$ by the word $y*x$.
```magma
> t := Substitute(w, p, #u, y*x);
> t;
y^-1 * x^-2 * y * x^3 * y
```
We create the set of all distinct cyclic permutations of the word $u$.
```magma
> rots := { RotateWord(u, i) : i in [1 ..#u] };
> rots;
{ y^-1 * x * y * z * x^-1 * z^-1, x * y * z * x^-1 * z^-1 * y^-1,
x^-1 * z^-1 * y^-1 * x * y * z, z * x^-1 * z^-1 * y^-1 * x * y,
z^-1 * y^-1 * x * y * z * x^-1, y * z * x^-1 * z^-1 * y^-1 * x }
```
### Finitely Generated Subgroups of Free Groups
A number of functions are available that apply specifically to finitely generated subgroups of free groups which, by the Nielsen–Schreier theorem, are themselves free. For example, it is possible to determine the index of one such subgroup in another, which may be infinite, and computation of intersections and normalisers of subgroups, and conjugacy testing are all possible. The relevant algorithms are described in [[Sim94](https://docs.magma-maths.org/references.md#cite-sims-fpgroups), Chapter 4], for example.
#### `Supergroup(F): GrpFP -> GrpFP`
The largest (with respect to containment) **free group** of which $F$ is known to be a subgroup. The remaining functions will fail and exit with errors if their arguments do not have a common supergroup.
#### `x in H: GrpFP, GrpFPElt -> BoolElt`
Returns `true` in $x \in H$, where $x$ is required to be an element of `Supergroup`(F).
#### `IsSubgroup(H, K): GrpFP, GrpFP -> BoolElt`
Given groups $H$ and $K$, subgroups of some common overgroup, return `true` if $H$ is a subgroup of $K$, and `false` otherwise.
#### `H eq K: GrpFP, GrpFP -> BoolElt`
Determines whether the (free) subgroups $H$ and $K$ of a common free group are equal.
#### `Index(F, H): GrpFP, GrpFP -> RngIntElt`
The index, which may be $\infty$, in the free group $F$ of subgroup $H$. An error will occur if $H$ is not a subgroup of $F$.
#### `HasFiniteIndex(F, H): GrpFP, GrpFP -> BoolElt`
Returns `true` if the index in the free group $F$ of subgroup $H$ is finite. An error will occur if $H$ is not a subgroup of $F$.
#### `FreeGenerators(H): GrpFP -> SeqEnum, GrpFP`
Returns a sequence of free generators of free group H and also the same group H defined on those generators.
#### `Rewrite(H, x): GrpFP, GrpFPElt -> BoolElt, GrpFPElt`
If $x$ is an element of $H$, return `true` and the corresponding word in the generators of $H$; otherwise return `false`.
#### `SubgroupElements(G, H, n): GrpFP, GrpFP, RngIntElt -> SetEnum`
Given a subgroup $H$ of a finitely presented group $G$, form the coset table for $H$ in $G$, and use it to return the *words* in the generators of $G$ of length at most $n$ that represent elements of $H$.
#### `H meet K: GrpFP, GrpFP -> GrpFP`
Computes the intersection of free groups $H$ and $K$.
#### `Centraliser(F, x): GrpFP, GrpFPElt -> GrpFP`
#### `Centralizer(F, x): GrpFP, GrpFPElt -> GrpFP`
Computes the centralizer in free group $F$ of element $x$. An error will occur if $x$ is not in $F$.
#### `IsConjugate(F, x, y): GrpFP, GrpFPElt, GrpFPElt -> BoolElt, GrpFPElt`
Determines whether elements $x$ and $y$ of free group $F$ are conjugate in $F$ and if so returns a conjugating element. An error will occur if either $x$ or $y$ is not in $F$.
#### `Centraliser(F, H): GrpFP, GrpFP -> GrpFP`
#### `Centralizer(F, H): GrpFP, GrpFP -> GrpFP`
Computes the centralizer in free group $F$ of subgroup $H$. An error will occur if $H$ is not a subgroup of $F$.
#### `Normaliser(F, H): GrpFP, GrpFP -> GrpFP`
#### `Normalizer(F, H): GrpFP, GrpFP -> GrpFP`
Computes the normalizer in free group $F$ of subgroup $H$. An error will occur if $H$ is not a subgroup of $F$.
#### `IsConjugate(F, H, K): GrpFP, GrpFP, GrpFP -> BoolElt, GrpFPElt`
Determines whether subgroups $H$ and $K$ of free group $F$ are conjugate in $F$ and if so returns a conjugating element. An error will occur if either $H$ or $K$ is not a subgroup of $F$.
#### `Example: Subgroups of a Free Group (ex-f8ece0)`
Starting with the free group of rank $3$ some subgroup calculations will be demonstrated.
```magma
> F := FreeGroup(3);
> F;
Finitely presented group F on 3 generators (free)
> H := sub< F | a, b>;
> H;
Finitely presented group H on 2 generators
Generators as words in group F
x = a
y = b
> Index(F, H);
0
```
The next step is to produce a more interesting subgroup of $F$. To get one, the subgroup $H$ will be conjugated by an element of $F$.
```magma
> K := H^(a*b*c);
> K;
Finitely presented group K on 2 generators
Generators as words in group F
K.1 = c^-1 * b^-1 * a * b * c
K.2 = c^-1 * b^-1 * a^-1 * b * a * b * c
> H eq K;
false
> H meet K;
Finitely presented group on 1 generator
Generators as words in group F
$.1 = Id(F)
> N := Normalizer(F, K);
> N;
Finitely presented group N on 2 generators
Generators as words in group F
N.1 = a^c
N.2 = b^c
> K eq N;
true
```
The normaliser $N$ of $K$ has nice generators but it is equal to $K$ so $K$ is self-normalising.
### The Automorphism Group of a Free Group
#### `AutomorphismGroup(F): GrpFP -> GrpAuto`
This intrinsic constructs the automorphism group of a free group $F$.
#### `InverseAutomorphismFreeGroup(F, Q): GrpFP, SeqEnum -> GrpAutoElt`
Given a free group $F$ and a sequence $Q$ of images of its generators, return the automorphism of $F$ defined by $Q$.
#### `WhiteheadReduction(F, Q): GrpFP, SeqEnum -> BoolElt, SeqEnum, GrpAutoElt`
Perform Whitehead reduction of a sequence $Q$ of elements of a free group $F$. Return true if the list $Q$ is part of a free basis, the Whitehead reduction, and corresponding automorphism of $F$.
#### `Example: Automorphism Group of a Free Group (ex-baab12)`
We find the automorphism group of the free group $F$ of rank $3$.
```magma
> F := FreeGroup(3);
> A, B := AutomorphismGroup(F);
> A;
A group of automorphisms of GrpFP: F
Generators:
Automorphism of GrpFP: F which maps:
F.1 |--> F.2
F.2 |--> F.1
F.3 |--> F.3
Automorphism of GrpFP: F which maps:
F.1 |--> F.2
F.2 |--> F.3
F.3 |--> F.1
Automorphism of GrpFP: F which maps:
F.1 |--> F.1^-1
F.2 |--> F.2
F.3 |--> F.3
Automorphism of GrpFP: F which maps:
F.1 |--> F.1 * F.2
F.2 |--> F.2
F.3 |--> F.3
> B;
Finitely presented group B on 4 generators
Relations
B.1^2 = Id(B)
B.2^3 = Id(B)
(B.2 * B.1)^2 = Id(B)
B.3^2 = Id(B)
B.3^-1 * B.1^-1 * B.2^-1 * B.3 * B.2 * B.1 = Id(B)
B.3^-1 * B.2^-1 * B.3^-1 * B.2 * B.3 * B.2^-1 * B.3 * B.2 = Id(B)
B.3^-1 * B.2^-1 * B.1^-1 * B.2 * B.3 * B.2^-1 * B.1 * B.2 = Id(B)
B.4^-1 * B.3^-1 * B.4^-1 * B.3^-1 * B.4 * B.3 * B.4 * B.3 = Id(B)
B.4^-1 * B.1^-1 * B.2^-1 * B.3^-1 * B.4^-1 * B.3^-1 * B.2 * B.1^-1 * B.4 *
B.1 * B.2^-1 * B.3 * B.4 * B.3 * B.2 * B.1 = Id(B)
(B.1 * B.3 * B.1 * B.4)^2 = Id(B)
B.4^-1 * B.1 * B.4 * B.1 * B.3 * B.4 * B.3 * B.1 * B.3 = Id(B)
B.4^-1 * B.2^-1 * B.1^-1 * B.2 * B.1^-1 * B.2^-1 * B.4 * B.2 * B.1 * B.2^-1
* B.1 * B.2 = Id(B)
B.4^-1 * B.2^-2 * B.3^-1 * B.2^2 * B.4 * B.2^-2 * B.3 * B.2^2 = Id(B)
B.4^-1 * B.1^-1 * B.2^-1 * B.1^-1 * B.2 * B.1^-1 * B.4^-1 * B.1^-1 * B.2^-1
* B.1^-1 * B.2 * B.1^-1 * B.4 * B.1 * B.2^-1 * B.1 * B.2 * B.1 * B.4 * B.1 *
B.2^-1 * B.1 * B.2 * B.1 = Id(B)
B.1 * B.2^-1 * B.4 * B.2 * B.1 * B.2^-1 * B.4 * B.2 * B.4 * B.2^-1 * B.4^-1
* B.2 * B.4^-1 = Id(B)
```
The first return value is the automomorphism group given as a group of mappings. The second return value is the automorphism group as an fp-group. A different type of group is needed in order to return the presentation of the automomorphism group.
#### `Example: Automorphism Group of a Free Subgroup (ex-1a5108)`
In this example the automorphism group of a subgroup of the free group $F$ of rank 3 is found.
```magma
> F := FreeGroup(3);
> H := sub< F | a, b>;
> K := H^(a*b*c);
> K;
Finitely presented group K on 2 generators
Generators as words in group F
K.1 = c^-1 * b^-1 * a * b * c
K.2 = c^-1 * b^-1 * a^-1 * b * a * b * c
> A, B := AutomorphismGroup(K);
> A;
A group of automorphisms of GrpFP: K
Generators:
Automorphism of GrpFP: K which maps:
K.1 |--> K.2
K.2 |--> K.1
Automorphism of GrpFP: K which maps:
K.1 |--> K.1^-1
K.2 |--> K.2
Automorphism of GrpFP: K which maps:
K.1 |--> K.1 * K.2
K.2 |--> K.2
> B;
Finitely presented group B on 3 generators
Relations
B.1^2 = Id(B)
B.2^2 = Id(B)
(B.2 * B.1)^4 = Id(B)
B.3^-1 * B.2^-1 * B.3^-1 * B.2^-1 * B.3 * B.2 * B.3 * B.2 = Id(B)
(B.1 * B.2 * B.1 * B.3)^2 = Id(B)
(B.3 * B.1 * B.2)^3 = Id(B)
```
## Introduction to Fp-Groups
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/introduction.md)
- [Overview](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/introduction.md#overview)
- [Definitions and Notation](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/introduction.md#definitions-and-notation)
- [Construction of FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md)
- [Relations](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#relations)
- [Quotient Group Constructors](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#quotient-group-constructors)
- [Example: Defining S4 as a Quotient Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-755b4c)
- [Example: Quotient Constructor without a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-17dd50)
- [FP-Group constructor](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#fp-group-constructor)
- [Example: Binary Tetrahedral Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-20a432)
- [Example: Quotient of an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-a5e77e)
- [Example: Coxeter's Parametrised Family of Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-17488a)
- [Presentation Operations](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#presentation-operations)
- [Example: Simplifying the Fibonacci Group F(8)](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-c32ec7)
- [Operations on Words](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#operations-on-words)
- [Presentations for Popular Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#presentations-for-popular-groups)
- [Example: Abelian Group as an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-318133)
- [Example: Symmetric Group Sym(8) as an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-169352)
- [Constructions for FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#constructions-for-fp-groups)
- [Example: Direct and Free Products of FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-7dc82f)
- [Example: FP-Group Isomorphic to GL(3,4)](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/fp-group-construction.md#example-ex-c105bf)
- [Properties of an FP-group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/Properties.md)
- [Example: Proving a Coxeter Group Infinite](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/Properties.md#example-ex-5d2db4)
- [Example: Automatic Structure of the Infinite Dihedral Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/Properties.md#example-ex-52e6c5)
- [Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md)
- [Subgroup Constructor](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#subgroup-constructor)
- [Example: Subgroup of Index 448](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-7c2095)
- [Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#coset-enumeration)
- [Example: Coset Table of an Index 15 Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-476bb1)
- [Coset Action](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#coset-action)
- [Example: Permutation Representation of G2(3)](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-6c4e10)
- [Enumeration of FI-Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#enumeration-of-fi-subgroups)
- [Example: Tutte's 8-Cage and the Conder Graph](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-c69596)
- [Example: Conjugacy Classes via Low Index Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-c3d6e8)
- [Operations for FI-Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#operations-for-fi-subgroups)
- [Example: Operations on a Subgroup of Index 448](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-f46c39)
- [Properties of Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#properties-of-subgroups)
- [Example: Subgroup Predicates in a Space Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-b1118e)
- [Presentations for Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#presentations-for-subgroups)
- [Example: Presentation of an Index 3 Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-d4d467)
- [Example: Rewriting a Presentation of L2(7)](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/subgroup-general.md#example-ex-282060)
- [Finite FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/Finite.md)
- [Example: Order of a Finite FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/Finite.md#example-ex-fb0cc3)
- [Example: Cayley Graph of a Finite FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/Finite.md#example-ex-19bd36)
- [Homomorphisms of FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md)
- [Homomorphism Constructor](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#homomorphism-constructor)
- [Example: Symmetric Group as Image of a Braid Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#example-ex-1337d3)
- [Searching for Homomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#searching-for-homomorphisms)
- [Example: Homomorphism onto A5](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#example-ex-1803a5)
- [Searching for Isomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#searching-for-isomorphisms)
- [Example: Isomorphism Search for Hillman's Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#example-ex-cebd96)
- [Example: Isomorphism Search for Neumann's Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/homomorphism.md#example-ex-8e44ee)
- [Quotient Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md)
- [Abelian Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#abelian-quotient)
- [Example: Structure of the Fibonacci Group F(7)](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-12a8db)
- [Example: Abelian Quotient Invariants of a Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-26e129)
- [$p$-Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#p-quotient)
- [Example: Largest 2-Quotient of Class 6](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-81ff16)
- [Example: 3-Quotient of a Group of Exponent 9](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-94855c)
- [Example: Metabelian 5-Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-e48428)
- [Nilpotent Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#nilpotent-quotient)
- [Example: Class 2 Nilpotent Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-0b5725)
- [Example: Free Nilpotent Groups via Nilpotent Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-470660)
- [Example: Nilpotent Quotient of a Baumslag-Solitar Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-f4b49e)
- [Soluble Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#soluble-quotient)
- [Example: Soluble Quotient of a Two-Generator Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-211a7e)
- [Example: Order of a Large Soluble Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-820eb8)
- [Simple Group Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#simple-group-quotients)
- [Example: Simple Quotients of a Perfect Group](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-af2186)
- [${\rm L}_2$-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#rm-l-2-quotients)
- [Example: L2-Quotients of FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-5313fb)
- [Example: Groups with Infinitely Many L2-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-0eaade)
- [Example: Matrix Groups of an L2-Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/quotient.md#example-ex-ba85d0)
- [Bibliography](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/bibliography.md)
### Introduction
#### Overview
Finitely presented groups (henceforth referred to as fp-groups) arise in many branches of mathematics and computation plays a significant role in their study. Unfortunately, answering most basic questions about a given fp-group is often computationally difficult. Over the past 50 years a range of computational tools for investigating fp-groups have been developed and most of these are installed in Magma. Some are straightforward to use but others are complicated.
The *word problem* for fp-groups asks whether there is an algorithm which given any element of an fp-group can decide whether or not it is the identity. In 1955, Novikov showed that there are specific fp-groups for which the word problem is insoluble. Further results show that other decision problems for fp-groups are also insoluble. Consequently, this limits what we can compute for fp-groups. The algorithms (or procedures) that have been developed can provide some information about many fp-groups but there will be many more groups for which they fail.
The Handbook documentation describing the fp-group facilities runs to more than 200 pages. In order to make the material more accessible this chapter is a terse introduction to the more important tools. Less important intrinsics are omitted and advanced features of those intrinsics that are discussed are often also omitted. The hope is that new users will be able to quickly get enough information from this chapter to be able to make straightforward application of the more important intrinsics and then dip into the detailed expositions in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp) when they become more proficient.
This chapter is concerned with groups defined by general presentations. Before starting this chapter the user should read the (very short) chapter [Free Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/index-free-groups.md#chapgrpfree) on free groups which is assumed knowledge for this chapter. For groups defined by power-conjugate presentations the user should consult chapters [Finite Soluble Groups](https://docs.magma-maths.org/FiniteGroups/FiniteSolubleGroups/index-finite-soluble-groups.md#chapgrppc) and [Polycyclic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/PolycyclicGroups/index-polycyclic-groups.md#chapgrpgpc). For users interested in automatic and hyperbolic groups most of the information concerning them can be found in chapter [Automatic and Hyperbolic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AutomaticAndHyperbolicGroups/index-automatic-and-hyperbolic-groups.md#chapgrpatc). The complete list of intrinsics for fp-groups together with their advanced features can be found in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
The intrinsics considered here are designed for doing what is sometimes referred to as *combinatorial group theory*. A description of the fundamental algorithms for finitely presented groups can be found in Sims [[Sim94](https://docs.magma-maths.org/references.md#cite-sims-fpgroups)] or Holt [[HEO05](https://docs.magma-maths.org/references.md#cite-holt-handbook)]. The facilities provided for fp-groups in Magma fall into the following categories:-
- Construction of presentations for fp-groups and their simplification.
- Determination of various properties such as non-trivial, finite/infinite, perfect, small cancellation, large, automatic, and hyperbolic.
- Calculations with subgroups of finite index including the enumeration of all subgroups having index less a specified bound.
- Construction of particular types of quotient group: abelian quotient, $p$-quotient, nilpotent quotient, soluble quotient, and simple group quotients.
The corresponding Magma categories are `GrpFP` for fp-groups and `GrpFPElt` for their elements.
#### Definitions and Notation
Let $X$ be a finite set of symbols, called generators in this context. Strings can be formed from $X$ by concatenating the symbols $x$ and $x^{-1}$ for $x \in X$. Defining multiplication $(*)$ of strings over $X$ as concatenation and forcing this multiplication rule to satisfy the group axioms puts a group structure on strings over $X$. In this group the identity is the empty string and the elements are called *words*. A group of this type is called the *free group* of rank $n$, where $n$ is the cardinality of $X$. These groups are discussed in detail in chapter [Free Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/index-free-groups.md#chapgrpfree).
A *relation* is an equality between two words while a *relator* consists of a word that equals the identity, that is, it is a relation where the right hand side is understood to be the identity. If a set $R$ of non-trivial relations are imposed on $F$ its elements are partitioned into equivalence classes. This set of equivalence classes form a new group. If $X$ and $R$ are both finite the group they define is called a *finitely-presented group* (abbreviated to fp-group) and is the subject of this chapter. The pair $\langle X | R \rangle$ is called a *presentation* for $G$. Note that any free group of finite rank is also an fp-group where the set $R$ is empty.
The set of relations $R$ can be converted into a set of relators and the relator words generate a subgroup $H$ of $F$. If $N$ is the normal closure of $H$ in $F$ then the group $G$ is the quotient group $F/N$. This observation is the basis for the way in which fp-groups are constructed in Magma. The free group of rank $n$ is constructed by the intrinsic `FreeGroup`$(n)$. Having constructed an appropriate free group $F$, an fp-group $G$ is constructed by specifying relations on the generators of $F$ that must be satisfied by the corresponding generators of $G$. The details of how this is achieved are discussed in the following sections.
### Construction of FP-Groups
An fp-group is normally either constructed by giving a presentation in terms of generators and relations or it is defined to be a subgroup or quotient group of an existing fp-group. However, fp-groups may be also created from finite permutation and matrix groups. Additionally, Magma has separate types for a number of families of fp-groups such as soluble groups which can be defined by power-conjugate presentations thus providing a solution to the word problem. Functions are provided to convert a member of one of these families into a general fp-group.
A finitely presented group $G$ is a quotient group of some free group having finite rank. The free group of rank $n$ is constructed by the intrinsic `FreeGroup`$(n)$. Having constructed an appropriate free group $F$, an fp-group $G$ is constructed by specifying relations on the generators of $F$ that must be satisfied by the corresponding generators of $G$. The details of how this is achieved are discussed in the following sections.
#### Relations
A *relation* is an equality between two words in an fp-group while a *relator* consists of a word that equals the identity, that is, it is a relation where the right hand side is understood to be the identity. A *presentation* is a set of symbols called generators together with a set of relations on those generators. To facilitate working with relations, a *relation type* is provided so that relations can be treated in a similar way to any other object in Magma. Below are the three operations that are specific to relations.
**Intrinsics:**
`w_1 = w_2` : Given words $w_1$ and $w_2$ belonging to fp-group $G$ create the relation $w_1 = w_2$.
`LHS`$(r)$ : Given a relation $r$ over the generators of $G$, return the left hand side of the relation $r$ as a word in $G$.
`RHS`$(r)$ : Given a relation $r$ over the generators of $G$, return the right hand side of the relation $r$ as a word in $G$.
#### Quotient Group Constructors
A group with non-trivial relations is constructed as a quotient of an existing group, usually a free group. For convenience, the necessary free group may be constructed in-line.
**Constructors:**
`quo`$\langle\ F\ |\ R\ \rangle$ : Given an fp-group $F$, and a set $R$ of relations and/or relators in the generators of $F$, construct the quotient $G$ of $F$ by the normal subgroup of $F$ defined by $R$. The group $G$ is defined by means of a presentation which consists of the relations for $F$ (if any), together with the additional relations defined by the list $R$. The constructor returns the quotient group $G$ and the natural homomorphism $\phi : F \rightarrow G$.
$G\ /\ H$ : Given a subgroup $H$ of the fp-group $G$, construct the quotient of $G$ by the normal closure $N$ of $H$. The quotient is formed by taking the presentation for $G$ and including the generating words of $H$ as additional relators.
**Notes:**
**(i)**
If explicit names are not available, a standard indexing notation is provided for referencing the generators of any group $G$. Thus, `G.i` denotes the $i$-th generator of $G$.
**(ii)**
Individual names may be given to the generators by means of the *generator-assignment* syntax. Suppose that the group $G$ is defined on $r$ generators. Then if $E$ is an expression creating $G$ the special assignment
$G\ \langle\ v_1,\ v_2,\ \ldots,\ v_r\ \rangle\ := E$;
is equivalent to the statements
$G := E;\ v_1 := G.1;\ v_2 := G.2;\ \dots;\ v_r := G.r$;
##### `Example: Defining S4 as a Quotient Group (ex-755b4c)`
Three variations of the constructor are used to define the fp-group $G$ isomorphic to the symmetric group of degree 4 with presentation $< a, b \ | \ a^2, b^3, (ab)^4 >$. Note that as $F$ (the free group) and $G$ (the $S_4$-quotient) are two distinct groups it is necessary to have different names for the generators of each. The relations given in the `quo` expression are relations on the free group generators.
```magma
> F := FreeGroup(2);
> G, phi := quo< F | a^2, b^3, (a*b)^4 >;
Finitely presented group G on 2 generators
Relations
x^2 = Id(G)
y^3 = Id(G)
(x * y)^4 = Id(G)
```
In the next example, the relations are given as a relations list.
```magma
> F := FreeGroup(2);
> G, phi := quo< F | a^2 = b^3 = (a*b)^4 = 1>;
> G;
Finitely presented group G on 2 generators
Relations
x^2 = Id(G)
y^3 = Id(G)
(x * y)^4 = Id(G)
```
Another alternative is to provide a set of relators:
```magma
> F := FreeGroup(2);
> rels := { a^2, b^3, (a*b)^4 };
> G, phi := quo< F | rels >;
> G;
Finitely presented group G on 2 generators
Relations
x^2 = Id(G)
(x * y)^4 = Id(G)
y^3 = Id(G)
```
The relations printed out are in a different order than in the set `rels` because sets are not ordered. If it is desired to have the relations appear in some particular order then a sequence should be used.
##### `Example: Quotient Constructor without a Free Group (ex-17dd50)`
A group may be defined using the `quo`-constructor without first assigning a free group. The \$ symbol is used to reference a group whose quotient is being formed. Recalling the indexing notation for generators, the generic names \$.1, \$.2, etc can be used in this situation.
```magma
> S4 := quo< FreeGroup(2) | $.1^2, $.2^3, ($.1*$.2)^4 >;
> S4;
Finitely presented group S4 on 2 generators
Relations
x^2 = Id(S4)
y^3 = Id(S4)
(x * y)^4 = Id(S4)
```
#### FP-Group constructor
For convenience, a constructor is provided which allows the user to define an fp-group without explicitly constructing a free group.
**Intrinsics:**
`FPGroup`$\langle\ X\ |\ R\ \rangle~:$ Given a list $X$ of variables $x_1, \ldots, x_r$, and a list of relations $R$ over these generators, first construct the free group $F$ on the generators $x_1, \cdots, x_r$ and then construct the quotient of $F$ corresponding to the normal subgroup of $F$ defined by the relations $R$. The syntax for the relations $R$ is the same as for the `quo`-constructor. The intrinsic returns the quotient group $G$ and the natural homomorphism $\phi : F \rightarrow G$.
##### `Example: Binary Tetrahedral Group (ex-20a432)`
The `FPGroup`-constructor is used to create the binary tetrahedral group using the presentation $< r, s \ | \ r^3 = s^3 = (rs)^2 >:$
```magma
> G := FPGroup< r, s | r^3 = s^3 = (r*s)^2 >;
> G;
Finitely presented group G on 2 generators
Relations
r^3 = (r * s)^2
s^3 = (r * s)^2
```
Again, using the `FPGroup`-constructor, the group $< r, s, t \ | \ r^2, s^2, t^2, rst = str = trs >$ would be specified as:
```magma
> G := FPGroup;
> G;
Finitely presented group G on 3 generators
Relations
r^2 = Id(G)
s^2 = Id(G)
t^2 = Id(G)
r * s * t = t * r * s
s * t * r = t * r * s
```
##### `Example: Quotient of an FP-Group (ex-a5e77e)`
As shown in this example, the [`quo< F | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)-constructor is not restricted to defining the quotient of a free group.
```magma
> F := FPGroup< x, y | x^2 = y^3 = (x*y)^7 = 1 >;
> F;
Finitely presented group F on 2 generators
Relations
x^2 = Id(F)
y^3 = Id(F)
(x * y)^7 = Id(F)
> G := quo< F | (x, y)^8 >;
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^3 = Id(G)
(a * b)^7 = Id(G)
(a, b)^8 = Id(G)
> #G;
10752
```
##### `Example: Coxeter's Parametrised Family of Groups (ex-17488a)`
In this example Magma functions are used to represent parametrised families of groups. In the notation of Coxeter, the symbol $(l, m | n, k)$ denotes the family of groups having presentation
$$
< a, b\ |\ a^l, b^m, (a*b)^n, (a*b^{-1})^k >.
$$
```magma
> Glmnk := func< l, m, n, k | FPGroup< a, b | a^l, b^m, (a*b)^n, (a*b^-1)^k > >;
> G := Glmnk(3, 3, 4, 4);
> G;
Finitely presented group G on 2 generators
Relations
a^3 = Id(G)
b^3 = Id(G)
(a * b)^4 = Id(G)
(a * b^-1)^4 = Id(G)
> Order(G);
168
> G := Glmnk(2, 3, 4, 5);
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^3 = Id(G)
(a * b)^4 = Id(G)
(a * b^-1)^5 = Id(G)
> Order(G);
1
```
Thus $(2, 3 \;\;|\;\; 4, 5 )$ is the trivial group.
#### Presentation Operations
**Intrinsics:**
`AddGenerator` $(G)$ : Add a new generator to the presentation defining $G$.
`AddGenerator` $(G,~w)$ : Add a new generator $z$ and the relation $z = w$ to the presentation defining $G$.
`AddRelation` $(G,~r)$ : Add the relation $r$ to the presentation defining $G$.
`AddRelation` $(G,~r,~i)$ : Add the relation $r$ after the $i$-th relation of the presentation defining $G$.
`DeleteGenerator` $(G,~x)$ : Delete the generator $x$ and all relations involving $x$ from the presentation defining $G$.
`ReplaceRelation` $(G,~s,~r)$ : Here each of $r$ and $s$ may be a relation or a word. The relation $r$ must be part of the presentation for $G$. The relation $r$ is replaced by the relation $s$ in the presentation defining $G$.
`DeleteRelation` $(G, ~r)$ : Delete the relation $r$ from the presentation defining group $G$.
`Generators` $(G)$ : The set of generators for $G$.
`NumberOfGenerators` $(G)$ : The number of generators used to define $G$.
`PresentationLength` $(G)$: The length of the presentation defining $G$. The length is defined to be the sum of the lengths of the words defining each relation or relator.
`Simplify`$(G)$ : A simplified presentation for $G$ is found by using Tietze transformations and substring searching. A new group $K$ defined by the simplified presentation is returned together with the isomorphism $f:G\rightarrow K$.
`SimplifyLength`$(G)$ : This has the same effect as `Simplify` except that the simplification process terminates when the presentation length starts to increase. While it may be possible to eliminate further generators it is likely that this will significantly lengthen the simplified presentation.
**Notes:**
**(i)**
The intrinsic `Simplify` works by repeatedly eliminating generators and subsequently shortening relators by substitution of substrings that correspond to the left or right hand side of a relation. The transformation process terminates when no more eliminations of generators and no more length reducing substring replacements are possible. The choice of simplification strategy can either be left to Magma or selected by the user, The simplification process can be controlled by a set of parameters described in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
##### `Example: Simplifying the Fibonacci Group F(8) (ex-c32ec7)`
Consider the Fibonacci group $F(8)$.
```magma
> F := FreeGroup(8);
> F8 :=
> quo< F | x1*x2=x3, x2*x3=x4, x3*x4=x5, x4*x5=x6,
> x5*x6=x7, x6*x7=x8, x7*x8=x1, x8*x1=x2>;
```
The intrinsic `Simplify` will be applied to obtain a presentation of $F(8)$ on fewer generators. It is returned as the presentation for a new group $H$. As the number of generators used in the simplified presentation for $H$ is unknown it is necessary to use a variation of the generator assignment expression: `H<[y]>`. This will give the generators of $H$ the names `y[1], y[2], ...`. Here $y$ is any variable name.
```magma
> H<[y]>, f := Simplify(F8);
> H;
Finitely presented group H on 2 generators
Generators as words in group F8
y[1] = x3
y[2] = x4
Relations
y[2] * y[1]^-2 * y[2] * y[1]^-1 * y[2]^2 * y[1] * y[2]^2 *
y[1]^-1 = Id(H)
y[1] * y[2] * y[1] * y[2]^2 * y[1] * y[2] * y[1]^2 * y[2]^-1
* y[1] = Id(H)
```
The isomorphism $f$ can be used to express the old generators in terms of the new ones.
```magma
> f;
Mapping from: GrpFP: F8 to GrpFP: H
> f(x1);
y[1]^2 * y[2]^-1
```
#### Operations on Words
The words $u$ and $v$ in the intrinsics below must belong to a common fp-group $G$.
**Intrinsics:**
$u\ *\ v$ : The product of words $u$ and $v$.
$u^{-1}$ : The inverse of word $u$.
`Identity` ($G)$ : The identity element for group $G$. Other notations are `Id` ($G)$ and $G!1$.
$u^n$ : The $n$-th power of $u$.
$u^v$ : The conjugate $v^{-1}*u*v$ of words $u$ and $v$.
$(u, v)$ : The commutator $u^{-1}*v^{-1}*u*v$ of words $u$ and $v$.
$u\ $`eq`$\ v$ : Returns `true` if the words $u$ and $v$ are identical when freely reduced. NB When $G$ is not a free group and `false` is returned, this does not imply that $u$ and $v$ do not represent the same element of $G$.
$u\ $`ne`$\ v$ : Returns `true` if the words $u$ and $v$ are distinct when freely reduced. NB When $G$ is not a free group and `true` is returned, this does not imply that $u$ and $v$ do not represent the same element of $G$.
#### Presentations for Popular Groups
Intrinsics are provided for constructing presentations for a number of familiar groups.
**Intrinsics:**
`AbelianFPGroup` $(Q)$ : The abelian group defined by the sequence of integers $Q$.
`CyclicFPGroup` $(n)$ : The cyclic group of order $n$.
`AlternatingFPGroup` $(n)$ : The alternating group of degree $n$ on two generators.
`BraidFPGroup` $(n)$ : The braid group on $n$ strings on $n-1$ Artin generators.
`DihedralFPGroup` $(n)$ : For $n > 2$, the dihedral group of order $2n$ on two generators.
`SymmetricFPGroup`$(n)$ : The symmetric group of degree $n$ on two generators.
##### `Example: Abelian Group as an FP-Group (ex-318133)`
An abelian group is presented as a direct product of cyclic groups. This direct product is defined by a sequence of integers giving the orders of the cyclic factors, where $0$ indicates ${\mathbb{Z}}$. To illustrate this the code below constructs the abelian group ${\mathbb{Z}}_4 \times {\mathbb{Z}}_9 \times {\mathbb{Z}}$.
```magma
> A := AbelianFPGroup([4, 9, 0]);
> A;
Finitely presented group A on 3 generators
Relations
a^4 = Id(A)
b^9 = Id(A)
a * b = b * a
a * c = c * a
b * c = c * b
```
##### `Example: Symmetric Group Sym(8) as an FP-Group (ex-169352)`
In this example the symmetric group ${\operatorname{Sym}}(8)$ is constructed as an fp-group:-
```magma
> S8 := SymmetricFPGroup(8);
> S8;
Finitely presented group S8 on 2 generators
Relations
x^8 = Id(S8)
y^2 = Id(S8)
(x * y)^7 = Id(S8)
(x^-1 * y * x * y)^3 = Id(S8)
(y * x^-2 * y * x^2)^2 = Id(S8)
(y * x^-3 * y * x^3)^2 = Id(S8)
(y * x^-4 * y * x^4)^2 = Id(S8)
```
#### Constructions for FP-Groups
Intrinsics are provided to construct products of fp-groups and also to transform groups defined by various types of polycyclic presentations into fp-groups. In addition there are intrinsics which construct a presentation for a finite permutation or matrix group $G$ and return a fp-group isomorphic to $G$.
**Intrinsics:**
`DirectProduct` $(G, H)$ : The direct product of the fp-groups $G$ and $H$.
`FreeProduct` $(G, H)$ : The free product of the fp-groups $G$ and $H$.
`FPGroup` $(T)$ : If $T$ is a finite nilpotent or soluble group defined by a polycyclic presentation return an fp-group $G$ isomorphic to $T$ together with the isomorphism $\phi: T \rightarrow G$.
`FPGroup` $(T)$ : If $T$ is a finite permutation or matrix group of moderate size then return an fp-group $G$ isomorphic to $T$ together with the isomorphism $\phi: G \rightarrow T$.
`FPGroupStrong` $(T)$ : If $T$ is a finite permutation or matrix group which may have order much greater than a million return an fp-group $G$ isomorphic to $T$. While `FPGroup` returns a presentation on the permutation/matrix generators, `FPGroupStrong` constructs a presentation on a larger set of generators (*strong generators* for $T$). An isomorphism $\phi: G \rightarrow T$ is also returned.
**Notes:**
**(i)**
Note that the intrinsic `FPGroup` applied to permutation or matrix groups may need to construct the coset table of $G$ over the identity and hence is limited to groups of order less than a million. The exact limit is determined by the amount of memory available.
##### `Example: Direct and Free Products of FP-Groups (ex-7dc82f)`
The direct product and free product of the alternating group of degree $7$ and the cyclic group of order $2$ are created as follows:-
```magma
> A7 := AlternatingFPGroup(7);
> A7;
Finitely presented group A7 on 2 generators
Relations
a^5 = Id(A7)
b^3 = Id(A7)
(a * b)^7 = Id(A7)
(b * a^-1 * b * a)^2 = Id(A7)
(b * a^-2 * b * a^2)^2 = Id(A7)
> Z2 := CyclicFPGroup(2);
Finitely presented group Z2 on 1 generator
Relations
c^2 = Id(Z2)
> G := DirectProduct(A7, Z2);
> G;
Finitely presented group G on 3 generators
Relations
x^5 = Id(G)
y^3 = Id(G)
(x * y)^7 = Id(G)
(y * x^-1 * y * x)^2 = Id(G)
(y * x^-2 * y * x^2)^2 = Id(G)
z^2 = Id(G)
x * z = z * x
y * z = z * y
> H := FreeProduct(A7, Z2);
> H;
Finitely presented group H on 3 generators
Relations
x^5 = Id(H)
y^3 = Id(H)
(x * y)^7 = Id(H)
(y * x^-1 * y * x)^2 = Id(H)
(y * x^-2 * y * x^2)^2 = Id(H)
z^2 = Id(H)
```
Note that the free product is always an infinite group.
##### `Example: FP-Group Isomorphic to GL(3,4) (ex-c105bf)`
In this example, an fp-group $G$ that is isomorphic to $\GL(3, 4)$ of order 181440 is created.
```magma
> k := GF(4);
> T := GL(3, k); #T;
181440
> T;
MatrixGroup(3, GF(2^2)) of order 2^6 * 3^4 * 5 * 7
Generators:
[ u 0 0]
[ 0 1 0]
[ 0 0 1]
[ 1 0 1]
[ 1 0 0]
[ 0 1 0]
> G := FPGroup(T);
> G;
Finitely presented group G on 2 generators
Relations
x^3 = Id(G)
y^7 = Id(G)
(x, y)^3 = Id(G)
(x*y^-1*x^-1*y^2)^3 = Id(G)
x^-1*y*x^-1*y^-1*x^-1*y*x^-1*y^-1*x*y*x*y^-1*x*y*x*y^-1 = Id(G)
y^-1*x^-1*y*x^-1*y^-1*x*y^-1*x^-1*y^-1*x*y*x*y^-1*x*y^3*x^-1 = Id(G)
(y^-1*x^-1*y^-1*x*y^-2)^3 = Id(G)
```
### Properties of an FP-group
Most of the group properties discussed in this section are usually hard to determine. Consequently, many of the intrinsics that test an fp-group $G$ for a given property $P$ only return a useful answer in those cases for which they are able to prove that $G$ has property $P$. If they cannot do this then no conclusion can be drawn. The group property intrinsics that are available in Magma are listed below:-
**Intrinsics:**
`IsPerfect` $(G)$ : Returns `true` if $G$ is perfect and `false` if not. Note that this and the small cancellation intrinsic are the only ones in this list that can definitively decide whether or not the condition holds.
`IsFinite` $(G)$ : Returns `true` if Magma is able to prove that $G$ has finite order. Otherwise no conclusion can be drawn. If successful the order of $G$ is also returned.
`IsInfinite` $(G)$ : Returns `true` if Magma is able to prove that $G$ is infinite. Otherwise no conclusion can be drawn.
`IsLarge` $(G, l, u)$ : Attempts to show that $G$ has a subgroup with homomorphic image the free group of rank 2. If successful, `true` is returned together with a subgroup witness to this property. If not successful, return `false`. The algorithm will consider subgroups of $G$ with index $h$ such that $l \le h \le u$.
`SmallCancellationConditions` $(G)$ : Returns the small cancellation conditions $T(i)$, $C(j)$, and $C'(k)$ that are satisfied by the presentation for $G$. Specifically it returns $i,\ j,\ k'$, where $T(i)$, $C(j)$, and $C'(k)$ are satisfied for all $k > k'$.
`IsAutomaticGroup` $(G)$ : Returns `true` if Magma is able to prove that $G$ is an automatic group. Otherwise no conclusion can be drawn. If successful the automatic structure for $G$ is also returned.
`IsHyperbolic` $(G)$ : Returns `true` if Magma is able to prove that $G$ is a hyperbolic group. Otherwise no conclusion can be drawn. If successful a Dehn function for the group may be returned.
**Notes:**
**(i)**
There are two basic approaches for computing structural information about an fp-group $G$: finding subgroups of finite index in $G$ where the index is moderate in size and finding quotients of $G$. These approaches are the basis for determining the first four properties above.
**(ii)**
Intrinsic `IsAutomatic` uses the Knuth–Bendix method for constructing a confluent presentation. The intrinsic works best for groups $G$ having two or three generators and short relators. Even so the running time can be substantial. Given the automatic structure for $G$, the word problem for $G$ can be solved, whether or not $G$ is finite or infinite can be easily determined, and a growth function for $G$ can be computed. See Chapter [Automatic and Hyperbolic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AutomaticAndHyperbolicGroups/index-automatic-and-hyperbolic-groups.md#chapgrpatc) for detailed information and examples.
**(iii)**
The algorithm underpinning the intrinsic `IsHyperbolic` is based on small cancellation theory. Care has to be taken in setting up the presentation and the user should read the relevant section in chapter [Automatic and Hyperbolic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AutomaticAndHyperbolicGroups/index-automatic-and-hyperbolic-groups.md#chapgrpatc) before attempting to use the intrinsic. As distinct from many of the other intrinsics described in this section, `IsHyperbolic` reaches a decision quite quickly. Preparing a presentation for input to this intrinsic is a challenge and the reader should consult the section on hyperbolic groups in chapter [Automatic and Hyperbolic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AutomaticAndHyperbolicGroups/index-automatic-and-hyperbolic-groups.md#chapgrpatc) for detailed information and examples.
#### `Example: Proving a Coxeter Group Infinite (ex-5d2db4)`
Some of the above intrinsics are applied to the Coxeter group
$$
G = \langle\ x,\ y\ |\ x^2,\ y^3,\ (x * y)^7,\ (x, y)^9\ \rangle
$$
First a proof that the group is infinite is produced:-
```magma
> G := FPGroup< x, y | x^2, y^3, (x * y)^7, (x, y)^9 >;
> yes, proof := IsInfinite(G);
> yes;
true
```
So the group is infinite. The second return argument outlines a proof of this fact.
```magma
> proof;
! The following lines sketch a proof that the fp-group F is infinite:-
! F has the group PSL(2, 8) of order 504 as a quotient G (L2Quotient).
! Let k denote the rational field.
! Let M be the ext square of perm module of G as a kG-module. It has dim 36.
! Let N be the kF-module given by the induced action of F on M.
! The dimension of the first integral cohomology group of the kF-module N is 1.
! As H^1(F, N) is non-zero, the Holt-Plesken criterion shows that F is infinite.
```
Next the group is tested for being perfect.
```magma
> IsPerfect(G);
true
```
So $G$ is perfect. Next a subgroup of finite index in $G$ is defined and tested for being perfect.
```magma
> H := sub< G | x, y * x * y * x * y * x * y^-1 >;
> Index(G, H);
9
> IsPerfect(H);
false
> AbelianQuotient(H);
Abelian Group isomorphic to Z/7
Defined on 1 generator
Relations:
7*$.1 = 0
```
So the subgroup $H$ is not perfect and its derived subgroup has index $7$ in $H$. Note that if $H$ had infinite or very large index in $G$ then the `IsPerfect` test would fail.
The final step is to determine the small cancellation conditions satisfied by $G$.
```magma
> a := SmallCancellationConditions(G);
This presentation satisfies T( 3 ).
This presentation satisfies C( 2 ).
This presentation satisfies C'(k) for all k > 1/2.
```
#### `Example: Automatic Structure of the Infinite Dihedral Group (ex-52e6c5)`
The infinite dihedral group is defined and then its automatic structure is computed.
```magma
> G := FPGroup< a, b | a^2, b^a = b^-1 >;
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^a = b^-1
> yes, A := IsAutomaticGroup(G);
> yes;
true
```
The second return value $A$ is the group defined by the automatic structure of $G$. The growth function $f$ for $A$ is calculated. The fact that the group is infinite means that the growth function has to be a rational function rather than a polynomial. Consequently, it is necessary to compute the power series for the rational function $f$.
```magma
> R := RationalFunctionField(Integers());
> f := GrowthFunction(A);
> f;
(-x^2 - 2*x - 1)/(x - 1)
```
This is a nice simple rational function. For a typical group the growth function has much larger degree numerator and denominator! The power series expansion of $f$ is obtained by coercing $f$ into the appropriate power series ring.
```magma
> PSR := PowerSeriesRing(Integers(): Precision := 15);
> s := PSR!f;
> s;
1 + 3*t + 4*t^2 + 4*t^3 + 4*t^4 + 4*t^5 + 4*t^6 + 4*t^7 + 4*t^8 +
4*t^9 + 4*t^10 + 4*t^11 + 4*t^12 + 4*t^13 + 4*t^14 + O(t^15)
```
So there are three elements of $G$ of length 1 and four elements of length 2.
### Subgroups
#### Subgroup Constructor
**Constructor:**
**sub**$\langle\ G\ |\ L\ \rangle ~$: Construct the subgroup $H$ of the fp-group $G$ generated by the words specified by the terms of the *generator list* $L = L_1, \ldots, L_r$. A term $L_i$ of the generator list may consist of a word, a set or sequence of words, a subgroup or a set or sequence of subgroups. The collection of words and groups specified by the list must all belong to the group $G$ and $H$ will be constructed as a subgroup of $G$.
**Notes:**
**(i)**
The generators of $H$ consist of the words specified directly by terms $L_i$ together with the stored generating words for any groups specified by terms of $L_i$. Repetitions of an element and occurrences of the identity element are removed (unless $H$ is trivial). If the `sub`-constructor is invoked with an empty list $L$, the trivial subgroup will be constructed.
##### `Example: Subgroup of Index 448 (ex-7c2095)`
The group $(8, 7\ |\ 2, 3)$ is defined by the presentation
$$
< a, b \ | \ a^8, b^7, (ab)^2, (ab^{-1})^3 >,
$$
and has a subgroup of index 448 generated by the words $a^2$ and $ab^{-1}$:
```magma
> G := FPGroup;
> G;
Finitely presented group G on 2 generators
Relations
a^8 = Id(G)
b^7 = Id(G)
(a * b)^2 = Id(G)
(a * b^-1)^3 = Id(G)
> H := sub< G | a^2, a * b^-1 >;
> H;
Finitely presented group H on 2 generators
Generators as words in group G
x = a^2
y = a * b^-1
```
#### Coset Enumeration
**Intrinsics:**
`Index` $(G,~H)$ : The index of $H$ in $G$.
`CosetTable` $(G,~H)$ : The coset table for $H$ in $G$ (right action).
`Transversal` $(G,~H)$ : A (right) Schreier system of coset representatives for $H$ in $G$ together with the coset map.
`SchreierGenerators` $(G,~H)$ : Schreier generators for $H$ as a set of words in $G$.
`ToddCoxeter` $(G, H)$ : The Todd–Coxeter procedure for coset enumeration is applied to construct the coset table for the subgroup $H$ of fp-group $G$. The convention here is that $G$ acts on the right. If successful the index of $H$ in $G$ and the coset table are returned otherwise zero is returned for the index.
`SetGlobalTCParameters` (~: `parameters`) : Set the parameter values used for indirect invocations of the Todd-Coxeter coset enumeration procedure.
`UnsetGlobalTCParameters` (~) : Restore the default values for the parameters used for indirect invocations of the Todd–Coxeter coset enumeration procedure.
**Notes:**
**(i)**
While strictly speaking the user does not need to invoke `ToddCoxeter` explicitly, it underpins many of the algorithms used to compute with subgroups in fp-groups and so a serious user needs to understand how to set the parameters in the case of computations that involve hard coset enumerations.
**(ii)**
The `ToddCoxeter` intrinsic has a large number of parameters which provide the user with control over the enumeration. Descriptions of these may be found in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp). Perhaps the most useful are `CosetLimit` which specifies the number of cosets which may be defined before abandoning an enumeration, `Time` which limits the maximum CPU time that can be expended, and `Strategy` which if set to ``` ``Hard" ``` chooses a strategy which has the best chance of succeeding with a hard enumeration.
**(iii)**
Two intrinsics, `SetGlobalTCParameters` and `UnsetGlobalTCParameters`, allow a user to set the parameters for implicit calls to the Todd–Coxeter algorithm.
**(iv)**
Descriptions of the Todd-Coxeter algorithm can be found in Chapter 5 of Holt [[HEO05](https://docs.magma-maths.org/references.md#cite-holt-handbook)] and Chapter 5 of Sims [[Sim94](https://docs.magma-maths.org/references.md#cite-sims-fpgroups)].
##### `Example: Coset Table of an Index 15 Subgroup (ex-476bb1)`
These intrinsics are illustrated using the infinite group $G$ defined by
$$
x^2 = y^3 = (xy)^7 = 1
$$
and $H$ a certain subgroup of $G$ having index 15.
```magma
> G := FPGroup< x, y | x^2, y^3, (x*y)^7 >;
> G;
Finitely presented group G on 2 generators
Relations
x^2 = Id(G)
y^3 = Id(G)
(x * y)^7 = Id(G)
>
> H := sub< G | x, y*x*y*x*y^-1*x*y^-1, y^-1*x*y^-1*x*y*x*y>;
> Index(G, H);
15
>
> CosetTable(G, H);
Mapping from: Cartesian Product<{ 1 .. 15 }, GrpFP: G> to { 1 .. 15 }
$1 $2 -$2
1. 1 2 3
2. 4 3 1
3. 5 1 2
4. 2 6 7
5. 3 8 9
6. 6 7 4
7. 10 4 6
8. 11 9 5
9. 9 5 8
10. 7 11 12
11. 8 12 10
12. 13 10 11
13. 12 14 15
14. 15 15 13
15. 14 13 14
>
> T := Transversal(G, H);
> T;
{@ Id(G), y, y^-1, y*x, y^-1*x, y*x*y, y*x*y^-1, x^y, y^-1*x*y^-1,
y*x*y^-1*x, y^-1*x*y*x, y*x*y^-1*x*y^-1, y*x*y^-1*x*y^-1*x,
y*x*y^-1*x*y^-1*x*y, y*x*y^-1*x*y^-1*x*y^-1 @}
```
#### Coset Action
The columns of the coset table for a subgroup $H$ in fp-group $G$ are permutations giving the action of the generators of $G$ on the cosets of $H$. If these columns are treated as permutation generators of a group $P$ then the homomorphism $\phi : G \rightarrow P$ is an epimorphism of $G$. This quotient group and its kernel often provide useful information about the structure of $G$. In Magma the group $G$ acts on the right of cosets and this should be assumed unless left action is explicitly stated.
**Intrinsics:**
`CosetAction` $(G, H)$ : The permutation representation $\phi$ of $G$ given by the action of $G$ on the cosets of $H$ in $G$. It returns $\phi$ and the image group $\phi(G)$. If the kernel of $\phi$ can be computed easily then it will also be returned.
`CosetImage` $(G, H)$ : The image of the permutation action of $G$ on the cosets of $H$.
`CosetKernel` $(G, H)$ : The kernel of the permutation action of $G$ on the cosets of $H$.
##### `Example: Permutation Representation of G2(3) (ex-6c4e10)`
The exceptional group $G_2(3)$ is a homomorph of the fp-group $G$ defined below. First, a permutation representation $G1$ on $351$ points is constructed for $G_2(3)$, and then the subgroup generated by the images of the first four generators of $G$ in $G1$ is computed. (The permutation group intrinsics are described in chapter [Permutation Groups](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm).)
```magma
> F := FreeGroup(7);
> z := y*c*a^2*b;
> u := x*d;
> t := w*c*a*d*b^2*c;
> G, psi :=
> quo< F | a^4, b^4, c^2, d^2, (a,b), (a*c)^2, (b*c)^2,
> (c*d)^2, d*a*d*b^-1, y^3, (a^-1*b)^y*d*a^-1*b^-1,
> (c*d*a^-1*b)^y*b^-1*a*d*c, a*d*y*d*a^-1*y, x^3,
> a^x*b^-1, b^x*a*b, c^x*c, (x*d)^2, (u*z)^6, w^3,
> (w,y), (a*b)^w*b^-1*a*d*c, (c*d*a^-1*b)^w*d*c*b^2,
> (b^2*c*d)^w*a^-1*b^-1, (c*a^2*b*w)^2,
> (a^-1*b)^w*d*a^-1*b^-1, (t*u)^3 >;
> z1 := psi(z);
> u1 := psi(u);
> t1 := psi(t);
> H := sub< G | z1*a^2*b^2, u1*c, t1*a^2*b^2 >;
> phi, G1, K := CosetAction(G, H);
> Degree(G1);
351
> print Order(G1), FactoredOrder(G1);
4245696 [ <2, 6>, <3, 6>, <7, 1>, <13, 1> ]
> CompositionFactors(G1);
G
| G(2, 3)
1
> S := sub< G1 | phi(a), phi(b), phi(c), phi(d) >;
> BSGS(S);
> S;
Permutation group S of degree 351
Order = 64 = 2^6
```
Thus the images of a, b, c, and d in G1 generate the Sylow $2$-subgroup of $G_2(3)$.
#### Enumeration of FI-Subgroups
If an fp-group $G$ has subgroups of finite index and their index is moderate then they may be enumerated. There are two methods, the first of which enumerates all subgroups down to a given index while the second enumerates normal subgroups down to a given index.
**Intrinsics:**
`LowIndexSubgroups` $(G, R)$ : Given a group $G$ (possibly the free group), and an expression $R$ defining a positive integer range (see below), determine the conjugacy classes of subgroups of $G$ whose indices lie in the range specified by $R$. The expression $R$ is either an integer $n$ which indicates the range $[1..n]$ or a tuple $\langle m,~n \rangle$ which represents the range $[m..n]$.
`LowIndexNormalSubgroups`$(G,~n)$ : The normal subgroups of $G$ having index in the range $[1..n]$ are enumerated. The subgroups are returned as a sequence of records (ordered by subgroup index). The upper bound for $n$ is $10^{10}$.
**Notes:**
**(i)**
The `LowIndexSubgroups` intrinsic has many applications and in favourable situations it can locate subgroups of index several hundred. The intrinsic also has parameters which can be found in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp). Two useful parameters are:
**-**
`Limit` which accepts a positive integer giving a limit on the number of subgroups satisfying expression $R$ that are sought.
**-**
`Print` which accepts an integer in the range $[0\ \ldots\ 3]$ which controls verbose output. Thus $0$ specifies no verbose output while the integers greater than $0$ turn on increasing amounts of verbose output.
**(ii)**
Another version of the `LowIndexSubgroups` intrinsic implements what is called a **process**. This version can be restarted and returns each subgroup immediately it is found. If more subgroups are needed then the computation is restarted. This is particularly useful in that a subgroup satisfying the user’s requirements may be found long before all the subgroups specified by $R$ are computed.
**(iii)**
The algorithm used to find subgroups of small index is described in Section 5.6 of Sims [[Sim94](https://docs.magma-maths.org/references.md#cite-sims-fpgroups)] and Section 5.4 of Holt [[HEO05](https://docs.magma-maths.org/references.md#cite-holt-handbook)].
##### `Example: Tutte's 8-Cage and the Conder Graph (ex-c69596)`
(Peter Lorimer) The two graphs known as Tutte’s 8-cage and the Conder graph may be constructed as the Cayley graphs of two conjugacy classes of subgroups having index $10$ in the finitely presented group
$$
\begin{aligned}.\end{aligned}
$$
The low index intrinsic will be used to construct these subgroups.
```magma
> G := FPGroup h^3 = a^2 = p^2 = 1, p^h = p, p^a = q,
> q^h = r, r^a = s, h^s = h^-1, r^h = p * q * r,
> s * r = p * q * r * s, p * q = q * p,
> p * r = r * p, p * s = s * p, q * r = r * q,
> q * s = s * q>;
> LowIndexSubgroups(G, <10, 10>);
[
Finitely presented group on 6 generators
Index in group G is 10 = 2 * 5
Generators as words in group G
$.1 = p
$.2 = s
$.3 = h
$.4 = q^-2
$.5 = a * h^-1 * a * r^-1
$.6 = a * h * a * h^-1 * a * q^-1,
Finitely presented group on 6 generators
Index in group G is 10 = 2 * 5
Generators as words in group G
$.1 = p
$.2 = s
$.3 = h
$.4 = q^-2
$.5 = a * h^-1 * a * r^-1
$.6 = a * h * a * h^-1 * a
]
```
##### `Example: Conjugacy Classes via Low Index Subgroups (ex-c3d6e8)`
A fairly surprising application for the low index subgroup algorithm is the enumeration of the conjugacy classes of subgroups of a finite fp-group. Consider the group
$$
{{\operatorname{PGL}}}_2(9) \simeq G =
\langle a,b | a^2,\ b^3,\ (ab)^8,\ [a,b]^5,\ [a,(ba)^3b^{-1}]^2 \rangle.
$$
```magma
> G := FPGroup< a,b | a^2, b^3, (a*b)^8, (a,b)^5,
> (a,(b*a)^3*b^-1)^2 >;
> Order(G);
720
```
In an infinite fp-group, finding all conjugacy classes of subgroups up to index 720 using the low index subgroup algorithm would be extremely difficult. In the case of a fp-group $G$ of order $720$, however, it is not so difficult.
```magma
> time sgG := LowIndexSubgroups(G, Order(G));
Time: 31.859
> #sgG;
26
```
A list of representatives of the 26 conjugacy classes of subgroups of $G$ have been found. For every representative, its index in $G$ and a set of generating words are stored. Here are two examples:-
```magma
> sgG[10];
Finitely presented group on 2 generators
Index in group G is 60 = 2^2 * 3 * 5
Generators as words in group G
$.1 = b
$.2 = a * b * a * b * a * b^-1 * a * b * a * b * a * b^-1 * a
* b^-1 * a
> sgG[21];
Finitely presented group on 1 generator
Index in group G is 180 = 2^2 * 3^2 * 5
Generators as words in group G
$.1 = (b * a)^2
```
#### Operations for FI-Subgroups
Operations on subgroups that construct a new subgroup or quotient group are listed. Here $G$ will be a finitely presented group with a presentation while $H$ and $K$ will be subgroups having finite index in $G$ unless otherwise stated. The subgroups $H$ and $K$ are assumed to be defined either by generators as words in $G$ or by the coset table for $H$ in $G$.
**Intrinsics:**
`H^u` : The conjugate $H^u$ where $u$ is a word in $G$.
$H$ `meet` $K$ : The intersection of subgroups $H$ and $K$ as a subgroup of $G$.
`G / H` : The quotient group of $G$ by the normal closure of subgroup $H$.
`Core` $(G,~H)$ : The core of $H$ in $G$.
`DerivedSubgroup` $(G)$ : The derived subgroup of $G$.
`TorsionFreeRank` $(G)$ : The torsion-free rank of the derived quotient group of $G$.
`MaximalOverGroup` $(G,~H)$ : A maximal overgroup of $H$ in $G$.
`MinimalOverGroup` $(G,~H)$ : A minimal overgroup of $H$ in $G$.
`NormalClosure` $(G,~H)$ : The normal closure of $H$ in $G$.
`Normaliser` $(G,~H)$ : The normaliser of $H$ in $G$.
**Notes:**
**(i)**
The first four intrinsics involve using the Todd–Coxeter algorithm. In the case of `DerivedSubgroup` it is necessary for Magma to be able to compute the coset table for the derived group of $G$. Thus the intrinsic will fail if the derived group has very large or infinite index in $G$.
**(ii)**
If $H$ is defined by a coset table then the `SchreierGenerators` intrinsic can be used to produce generating words for $H$. In many cases this will be done automatically.
##### `Example: Operations on a Subgroup of Index 448 (ex-f46c39)`
The group $(8, 7\ |\ 2, 3)$ is defined by the presentation
$$
< a, b \ | \ a^8, b^7, (ab)^2, (ab^{-1})^3 >,
$$
and has a subgroup of index 448 generated by the words $a^2$ and $ab^{-1}$:
```magma
> G := FPGroup;
> G;
Finitely presented group G on 2 generators
Relations
a^8 = Id(G)
b^7 = Id(G)
(a * b)^2 = Id(G)
(a * b^-1)^3 = Id(G)
> H := sub< G | a^2, a * b^-1 >;
> H;
Finitely presented group H on 2 generators
Generators as words in group G
x = a^2
y = a * b^-1
```
The subgroup functions will now be demonstrated on the subgroup $H$. First of all its index and order are calculated.
```magma
> Index(G, H);
448
> Order(H);
24
```
The derived group of $G$ is trivial:-
```magma
> DerivedSubgroup(G);
Finitely presented group
Index in group G is 1 = 1
Subgroup of group G defined by coset table
```
The group $G$ is perfect so it is necessary to look elsewhere to find some other subgroups.
```magma
> P := MinimalOvergroup(G, H);
> P;
Finitely presented group P on 3 generators
Index in group G is 224 = 2^5 * 7
Generators as words in group G
P.1 = a^2
P.2 = a * b^-1
P.3 = a * b^2 * a^3 * b^-1 * a^3 * b^3
> Order(P);
48
```
So $H$ sits inside a subgroup $P$ having twice its order.
```magma
> N := Normalizer(G, H);
> N;
Finitely presented group N on 4 generators
Index in group G is 112 = 2^4 * 7
Generators as words in group G
N.1 = a^2
N.2 = a * b^-1
N.3 = b^-1 * a * b^2 * a^4 * b^-1 * a * b^2
N.4 = a * b^2 * a^3 * b^-1 * a^3 * b^3
> Order(N);
96
```
The subgroup $H$ has a normaliser $N$ of order $96$.
```magma
> Q := MaximalOvergroup(G, H);
> Q;
Finitely presented group Q on 3 generators
Index in group G is 7
Generators as words in group G
Q.1 = a^2
Q.2 = a * b^-1
Q.3 = b^-1 * a * b^2
> #Q;
1536
```
So the largest subgroup $Q$ that contains $H$ has index $7$ in $G$.
```magma
> M := NormalClosure(G, H);
> M;
Finitely presented group M on 7 generators
Index in group G is 1 = 1
Generators as words in group G
M.1 = a^2
M.2 = a * b^-1
M.3 = a^2 * b^-1 * a^-1
M.4 = b^-1 * a
M.5 = b * a * b^-2
M.6 = (b^-1 * a * b)^2
M.7 = b^-1 * a
```
To summarise, proper subgroups of $G$ having orders 48, 96, and 1536 that contain $H$ have been found. The normal closure of $H$ is $G$.
#### Properties of Subgroups
The operations described in this subsection all require a closed coset table for at least one subgroup of an fp-group. If a closed coset table is needed and has not been computed, a coset enumeration will be invoked. If the coset enumeration does not produce a closed coset table, a runtime error is reported.
Experienced users can control the behaviour of such indirectly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, refer to chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
**Intrinsics:**
`u in H` : True if and only if the word $u$ in $G$ is a member of $H$.
`u notin H` : True if and only if the word $u$ in $G$ is not a member of $H$.
`H eq K` : True if and only if subgroups $H$ and $K$ are equal.
`H ne K` : True if and only if subgroups $H$ and $K$ are not equal.
`H subset K` : True if and only if the subgroup $H$ is a subset of $K$.
`IsConjugate`$(G, H, K)$ : True if and only if subgroups $H$ and $K$ are conjugate in $G$. If $H$ and $K$ are conjugate an element that conjugates $H$ into $K$ is also returned.
`IsMaximal`$(G, H)$ : True if subgroup $H$ is maximal in $G$.
`IsNormal`$(G, H)$ : True if subgroup $H$ is normal in $G$.
`IsSelfNormalizing`$(G, H)$ : True if subgroup $H$ is a self-normalising subgroup of $G$.
`IsPerfect`$(H)$ : True if the subgroup $H$ is perfect.
##### `Example: Subgroup Predicates in a Space Group (ex-b1118e)`
Some of these subgroup predicates are illustrated by applying them to some subgroups of the two-dimensional space group $p4g = < r,\ s\ |\ r^2,\ s^4,\ (r, s)^2 >$.
```magma
> p4g := FPGroup< r, s | r^2 = s^4 = (r,s)^2 = 1 >;
> h := sub< p4g | (s^-1*r)^4, s*r >;
> k := sub< p4g | (s^-1*r)^2, (s*r)^2 >;
```
Subgroups $h$ and $k$ have the same index in $p4g$ but they are neither equal nor conjugate.
```magma
> Index(p4g, h), Index(p4g, k);
8 8
> h eq k;
false
> IsConjugate( p4g, h, k);
false
```
Are subgroups $h$ and $k$ normal in $p4g$?
```magma
> IsNormal(p4g, h), IsNormal(p4g, k);
false true
```
Since $h$ is not normal it makes sense to construct its normal closure $n$ in $p4g$. It turns out that $n$ is maximal in $p4g$ and contains $k$.
```magma
> n := NormalClosure(p4g, h);
Index( p4g, n)l
2
> IsMaximal(p4g, n);
true
> k subset n;
true
```
Here is a new subgroup $l$ of $p4g$ that turns out to be conjugate to $h$.
```magma
> l := sub< p4g | (s*r)^4, s^-1*r >;
> IsConjugate(p4g, h, l);
true r^-1
```
i.e. $l = h^{r^{-1}}$. The intersection of $h$ and $h^{r^{-1}}$ already yields the core of $h$ in $p4g$.
```magma
> (h meet l) eq Core(p4g, h);
true
```
#### Presentations for Subgroups
Let $H$ be a subgroup of finite index in the finitely presented group $G$. It frequently happens that it is desirable to construct a set of defining relations for $H$ from those of $G$. Such a presentation can be obtained either on a set of Schreier generators for $H$ or on the given generators of $H$ using the Reidemeister–Schreier rewriting technique [[MKS76](https://docs.magma-maths.org/references.md#cite-magnus-cgt)], if necessary together with extended coset enumeration [[AR84](https://docs.magma-maths.org/references.md#cite-extended-tc), [HKRR84](https://docs.magma-maths.org/references.md#cite-havas-tietze)]. It should be emphasised that if the user wishes only to determine the structure of the maximal abelian quotient of $H$, then the function `AbelianQuotientInvariants` should be used. In this case there is no need to first construct a presentation for $H$ using the `Rewrite` function described below, since `AbelianQuotientInvariants` employs a special form of the Reidemeister–Schreier rewriting process which abelianises each relator as soon as it is constructed. Thus, compared to the function `Rewrite`, the function `AbelianQuotientInvariants` can be applied to subgroups of much larger index.
**Intrinsics:**
`Rewrite` $(G,~H)$ : Given an fp group $G$ and a subgroup $H$ having finite index in $G$, return a group $K$ isomorphic to $H$ with a presentation on some of the Schreier generators of $H$ in $G$. The group $K$ is created as a subgroup of $G$ with a presentation on the new generators. The isomorphism from $H$ onto $K$ is returned as second return value. Note that the generators of $K$ will, in general, not correspond to the generators of $H$.
`Rewrite` $(G,~\sim H)$ : Given a finitely presented group $G$ and a subgroup $H$ having finite index in $G$, a set of defining relations for $H$ is constructed on the given generators. The information stored for $H$ is changed to include these defining relations.
**Notes:**
**(i)**
The first version of `Rewrite` selects its own generators and consequently is faster and often produces a simpler presentation. The second version constructs a presentation on the given generators. This is a more complicated process and the resulting presentation is often large. Consequently, by default, simplification is attempted using Tietze transformations that do not involve any changes to the generators of $H$. If is not necessary to obtain a presentation for $H$ on the original generators of $H$, the first method is strongly recommended.
##### `Example: Presentation of an Index 3 Subgroup (ex-d4d467)`
Starting with the group $G$ defined by
$$
< x, y\ |\ x^2,\ y^3,\ (xy)^{12},\ (xy)^6(xy^{-1})^6 >,
$$
a subgroup $H$ of index $3$ is generated by the words $x$, $yxy^{-1}$, and $yxy^{-1}xy^{-1}xy$:
```magma
> G := FPGroup< x, y | x^2, y^3, (x*y)^12, (x*y)^6*(x*y^-1)^6 >;
> G;
Finitely presented group G on 2 generators
Relations
x^2 = Id(G)
y^3 = Id(G)
(x * y)^12 = Id(G)
x * y * x * y * x * y * x * y * x * y * x * y * x * y^-1 * x *
y^-1 * x * y^-1 * x * y^-1 * x * y^-1 * x * y^-1 = Id(G)
> H := sub< G | x, y*x*y^-1, y*x*y^-1*x*y^-1*x*y >;
> H;
Finitely presented group H on 3 generators
Generators as words in group G
H.1 = x
H.2 = y * x * y^-1
H.3 = y * x * y^-1 * x * y^-1 * x * y
> Index(G, H);
3
```
A presentation for $H$ is now computed. Then using `AbelianQuotientInvariants`, $H$ is found to have a non-trivial soluble quotient. Using the $p$-quotient algorithm described later, $H$ is found to have a 2-quotient of order$2^{62}$!
```magma
> K := Rewrite(G, H);
> K;
Finitely presented group K on 3 generators
Generators as words in group G
K.1 = x
K.2 = y * x * y^-1
K.3 = x^y
Relations
K.1^2 = Id(T)
K.2^2 = Id(T)
K.3^2 = Id(T)
(K.3 * K.2 * K.1 * K.3 * K.2)^2 = Id(K)
(K.1 * K.3 * K.2 * K.1 * K.3)^2 = Id(K)
(K.1 * K.2 * K.1 * K.3 * K.2)^2 = Id(K)
> AbelianQuotientInvariants(K);
[ 2, 2, 2 ]
> Q2 := pQuotient(K, 2, 30);
> FactoredOrder(Q2);
[ <2, 62> ]
```
##### `Example: Rewriting a Presentation of L2(7) (ex-282060)`
This example demonstrates how the intrinsic `Rewrite` can be used to obtain a presentation of a finitely presented group on a different set of generators.
Firstly, a presentation for $L_2(7)$ on two generators $x$ and $y$ is used to define the group.
```magma
> L27 := FPGroup< x, y | x^3 = 1, y^3 = 1, (x*y)^4 = 1,
> (y*y^x)^2 = y^x*y >;
```
The group is also generated by the elements $a = (xy)^2$ and $b = y$.
```magma
> H := sub;
> Index(L27, H);
1
```
At this point, no defining relations of $H\cong L27$ on the generators $a$ and $b$ are known.
```magma
> H;
Finitely presented group H on 2 generators
Index in group L27 is 1
Generators as words in group L27
a = (x * y)^2
b = y
```
The intrinsic `Rewrite` is applied to $H$ as a subgroup of $L27$ in order to compute defining relations on the generators $a$ and $b$.
```magma
> Rewrite(L27, ~H);
> H;
Finitely presented group H on 2 generators
Index in group L27 is 1
Generators as words in group L27
a = (x * y)^2
b = y
Relations
a^2 = Id(H)
b^3 = Id(H)
(a * b)^7 = Id(H)
(a * b^-1 * a * b)^4 = Id(H)
(b * a * b^-1 * a * b * a)^4 = Id(H)
```
The fifth relation turns out to be redundant: $a$ and $b$ are standard generators for $L_2(7)$ as is shown here:
```magma
> Order(DeleteRelation(H, 5)) eq Order(H);
true
```
### Finite FP-Groups
Finite fp-groups are often of interest and so a small number of intrinsics are provided for such groups.
**Intrinsics:**
`IsFinite` $(G)$ : Returns true if $G$ is proved to be finite, false if unknown. If $G$ is found to be finite, its order is also returned.
`Order` $(G)$ : Given a finite fp-group $G$ return the order of $G$. If the order cannot be determined or is found to be infinite, then the value zero or `Infinity`, respectively, is returned.
`PermutationGroup` $(G)$ : Given a finite fp-group $G$ return a permutation group $P$ that is isomorphic to $G$ together with the isomorphism $\phi: G \rightarrow P$.
`CayleyGraph` $(G)$ : If $G$ is an fp-group with generating set $X$ return the Cayley graph $CG$ of $G$ with respect to the generating set $X$. The Cayley graph is returned as a Magma directed graph with its vertices and edges labelled. A parameter `Labelled` can be set to `false` to prevent the edges and vertices being labelled.
**Notes:**
Note that the intrinsics `IsFinite`, `Order`, `PermutationGroup`, and `CayleyGraph` may need to construct the coset table of $G$ over the identity and hence are generally restricted to groups of order less than a million. The exact limit is determined by the amount of memory available.
#### `Example: Order of a Finite FP-Group (ex-fb0cc3)`
The intrinsic `Order` is invoked without any parameter settings to compute the order of the group
$$
G =\,\, \langle a, b\ |\ a^8, b^7, (ab)^2, (ab^{-1})^3 \rangle
$$
```magma
> G := FPGroup;
> G;
Finitely presented group G on 2 generators
Relations
a^8 = Id(G)
b^7 = Id(G)
(a * b)^2 = Id(G)
(a * b^-1)^3 = Id(G)
> Order(G);
10752
```
#### `Example: Cayley Graph of a Finite FP-Group (ex-19bd36)`
Now the Cayley graph of the group defined in the previous example is constructed. Next the greatest distance between any pair of vertices (the diameter) is found.
```magma
> G := FPGroup;
> CG := CayleyGraph(G);
> Diameter(CG);
22
```
Finally, something more sophisticated is demonstrated. First the shortest path $g$ (geodesic) between vertex 1 and vertex $1000$ is found. Then the labels on the edges of this path are multiplied together yielding the element of $G$ corresponding to vertex $1000$.
```magma
> V := Vertices(CG);
> E := Edges(CG);
> g := Geodesic(V!1, V!1000);
> g;
[ 1, 2, 6, 16, 38, 33, 73, 82, 163, 309, 168, 318, 579, 999, 1642, 1000 ]
> w := &*[ Label(E![g[i], g[i+1]]) : i in [1.. #g-1] ];
> w;
a^5 * b * a^3 * b^4 * a * b
```
### Homomorphisms of FP-Groups
In this section the following topics are covered:
**(i)**
The specialisation of the general `hom`-constructor to the case in which the domain is an fp-group
**(ii)**
Searching for homomorphisms from an fp-group to a given permutation group.
**(iii)**
Searching for an isomorphism between two fp-groups.
While this section presents general machinery for fp-group homomorphisms the following section covers more specialised tools, specifically those that search for specific types of epimorphisms.
#### Homomorphism Constructor
The specialisation of the general `hom`–constructor to the case in which the domain is an fp-group is described in this section.
**Constructor/Intrinsic:**
`hom`$\langle\ P \ar G\ |\ S \ \rangle$ : Returns the homomorphism from the $n$-generator fp-group $P$ to the group $G$ defined by the assignment $S$. Here $S$ is either a list, sequence, or indexed set containing the images of the $n$ generators of $P$ or a list, sequence, enumerated set, or indexed set, containing $n$ tuples $\langle x_i,\ y_i\rangle$ or $n$ arrow pairs $x_i \ar y_i$ defining the $n$ generator-image pairs.
`IsSatisfied`$(U, E)$ : Let $U$ be a set or sequence of words or relations constituting the defining presentation of an $n$-generator fp-group $H$ and $E$ be a sequence of $n$ elements in a group $G$ for which there is a unique representation for each element. This intrinsic evaluates the words/relations of $U$ in $G$ where the $i$-th generator of $H$ maps to the $i$-th element of $G$ and returns `true` if and only if each word in $U$ evaluates to the identity or each relation in $U$ is satisfied.
**Notes:**
**(i)**
The `hom` constructor does not check that the given mapping is a homomorphism and so the `IsSatisfied` intrinsic is provided to allow the user to verify the correctness of the definition of those homomorphisms that are from an fp-group to a group whose elements have unique representations.
**(ii)**
The kernel of a homomorphism having a domain of type `GrpFP` can be calculated using the intrinsic `Kernel` in many cases. This is the case if the codomain is one of the group types `GrpPerm`, `GrpMat`, `GrpAb`, `GrpPC`, `GrpGPC`, as well as the image being finite and its order small. The kernel may also be computable when the codomain is of the type `GrpFP` if the image is finite, sufficiently small, and has a known presentation.
**(iii)**
The standard tools for working with homomorphisms are available for fp-groups: image of an element or subgroup `x@phi` (or `phi(x)`, preimage of an element or subgroup `x@@phi`, domain (`Domain`), codomain (`Codomain`), image (`Image`), and kernel (`Kernel`). However, in some cases it will not be possible to compute the kernel of a homomorphism and if the kernel is not available then the computation of preimages of subgroups will not be possible.
##### `Example: Symmetric Group as Image of a Braid Group (ex-1337d3)`
For arbitrary $n>0$, the symmetric group of degree $n+1$ is an epimorphic image of the braid group on $n$ generators. In this example, this relationship for the case $n=4$ will be exhibited.
The first step is to create the braid group $B$ on 5 strings, that is, 4 Artin generators.
```magma
> B := BraidFPGroup(5);
> B;
Finitely presented group B on 4 generators
Relations
x * y * x = y * x * y
x * z = z * x
x * t = t * x
y * z * y = z * y * z
y * t = t * y
z * t * z = t * z * t
```
In the symmetric group of degree 5, 4 transpositions are defined which will be the images of the generators of $B$.
```magma
> S := Sym(5);
> imgs := [ S!(1,2), S!(2,3), S!(3,4), S!(4,5) ];
```
In order to verify that this assignment actually gives rise to a well-defined homomorphism, a check is made to decide whether the potential images satisfy the defining relations of $B$.
```magma
> rels := Relations(B);
> rels;
[ x*y*x = y*x*y, x*z = z*x, x*t = t*x, y*z*y = z*y*z,
y*t = t*y, z*t*z = t*z*t ]
> IsSatisfied(rels, imgs);
true
```
They do. So now the homomorphism from $B$ to $S$ can be defined:-
```magma
> f := hom< B->S | imgs >;
```
Note that $f$ is surjective, i.e., $S$ is an epimorphic image of $B$ as claimed above.
```magma
> f(B) eq S;
true
```
What is the kernel of $f$?
```magma
> Kernel(f);
Finitely presented group
Index in group B is 120 = 2^3 * 3 * 5
Subgroup of group B defined by coset table
```
Using the function `GeneratingWords` described elsewhere, a set of generators for $\ker(f)$ as a subgroup of $B$ can be constructed.
```magma
> GeneratingWords(B, Kernel(f));
{ x^-2, y^-2, z^-2, t^-2, (y*z^-1*y^-1)^2, (z*t^-1*z^-1)^2,
(x*y^-1*x^-1)^2, (x*y*z^-1*y^-1*x^-1)^2, (y*z*t^-1*z^-1*y^-1)^2,
(x*y*z*t^-1*z^-1*y^-1*x^-1)^2 }
```
It is easy to see that all generators of $\ker(f)$ are conjugates of words of the form $g^{\pm2}$, where $g$ is a generator of $B$. This is checked, using the normal closure intrinsic `NormalClosure`.
```magma
> N := sub< B | x^2, y^2, z^2, t^2 >;
> Kernel(f) eq NormalClosure(B, N);
true
```
Thus the braid relations together with the relators $x^2, y^2, z^2, t^2$ are a set of defining relations for $S$. This is confirmed by running the following code:-
```magma
> BS := quo< B | x^2, y^2, z^2, t^2 >;
> Order(BS);
120
> P := PermutationGroup(BS);
> P;
Permutation group P acting on a set of cardinality 10
Order = 120 = 2^3 * 3 * 5
> IsIsomorphic(P, Sym(5));
true
```
#### Searching for Homomorphisms
This section describes functions for computing representatives of the classes of homomorphisms from a finitely presented group $G$ to a finite group $H$ modulo a group $A$ of automorphisms of $H$. The algorithm and code are due to Derek Holt.
**Intrinsics:**
`Homomorphisms` $(G, H, A )$ : Given a finitely presented group $G$ and two permutation groups $H$ and $A$ with $H\triangleleft A$, a sequence containing representatives of the classes of homomorphisms from $G$ to $H$ modulo automorphisms of $G$ induced by elements of $A$ is returned.
`Homomorphisms` $(G, H, A )$ : This version is identical except that $H$ and $A$ are polycyclic groups rather than permutation groups.
**Notes:**
**(i)**
Two homomorphisms $\phi_1$, $\phi_2 : G \rightarrow H$ are considered equivalent if there exists an element $a\in A$ such that $\phi_1(x) = \phi_2(x)^a$ for all $x\in G$.
**(ii)**
There is a process version of this intrinsic; that is, a restartable version that returns each homomorphism as soon as it is discovered.
**(iii)**
The call `Homomorphisms(G, H, H)` may be abbreviated to `Homomorphisms(G, H)`.
**(iv)**
The permutation group version of the intrinsic has parameters `Surjective`, `Limit`, `TimeLimit`, `CosetEnumeration`, and `CacheCosetAction`. If `Surjective` is set to `true` then the search is restricted to epimorphisms. The final two parameters relate to the runtime and memory usage of the calculation. The polycyclic group version of the intrinsic has only parameters `Surjective` and `Limit`. The default values of `Surjective` and `Limit` are `true` and 1, respectively. If `Limit` is set to $0$ this indicates no limit.
##### `Example: Homomorphism onto A5 (ex-1803a5)`
Consider the finitely presented group
$$
G\ := \ < a,b,c \ |\ ac^{-1}bc^{-1}aba^{-1}b,
abab^{-1}c^2b^{-1},
a^2b^{-1}(ca)^4cb^{-1}>.
$$
```magma
> G := FPGroup< a,b,c | a*c^-1*b*c^-1*a*b*a^-1*b,
> a*b*a*b^-1*c^2*b^-1,
> a^2*b^-1*c*a*c*a*c*a*c*a*c*b^-1 >;
```
The intrinsic `Homomorphisms` will be used to prove that there is a homomorphism from $G$ onto $A_5$.
```magma
> H := Alt(5);
> homs := Homomorphisms(G, H : Limit := 1);
> homs;
[
Homomorphism of GrpFP: G into GrpPerm: H, Degree 5, Order 2^2 * 3 * 5
G.1 |--> (1, 2, 3)
G.2 |--> (1, 2, 4)
G.3 |--> (2, 4)(3, 5)
]
```
Hence there exists at least one such homomorphism. Are there any more?
```magma
> homs := Homomorphisms(G, H);
> #homs;
4
```
So there are four distinct classes of homomorphisms! Finally, the kernel of the first of these homomorphisms is found.
```magma
> phi := homs[1];
> K := Kernel(phi);
> K;
Finitely presented group K
Index in group G is 60 = 2^2 * 3 * 5
Subgroup of group G defined by coset table
```
#### Searching for Isomorphisms
This section describes an intrinsic that searches for isomorphisms between two finitely presented groups. The algorithm and code are due to Derek Holt.
**Intrinsics:**
`SearchForIsomorphism` $(G,~H,~m)$ : Attempt to find an isomorphism from the finitely presented group $G$ to the finitely presented group $H$. The search will be restricted to those homomorphisms for which the sum of the word-lengths of the images of the generators of $G$ in $H$ is at most $m$. If an isomorphism $\phi$ is found, then the values `true`, $\phi$, $\phi^{-1}$ are returned. Otherwise, the values `false`, \_, \_ are returned; of course, that does not necessarily mean that the groups are not isomorphic.
**Notes:**
**(i)**
Setting the verbose flag `"IsoSearch"` to $1$, will cause information about the progress of the search to be printed.
**(ii)**
There are a number of parameters available for the intrinsic `SearchForIsomorphism`. If the boolean parameter `All` is set to `true` then the search will be continued until all possible isomorphisms that are defined by images of the generators of $G$ in $H$ that satisfy the image length condition are considered. If the boolean parameter `IsomsOnly` is set to `false` then homomorphisms $\phi:F\to G$ will be sought. The reader should consult chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp) for a full description of all the parameters.
##### `Example: Isomorphism Search for Hillman's Groups (ex-cebd96)`
Jonathon Hillman asked whether the following two groups are isomorphic:
```magma
> G1 := FPGroup u*(s*t)^2=(s*t)^2*u >;
> G2 := FPGroup y^2=(z^-1*x)^2, z*(x*y)^2=(x*y)^2*z >;
> isiso, f1, f2 := SearchForIsomorphism(G1,G2,7);
> isiso;
true
> f1;
Homomorphism of GrpFP: G1 into GrpFP: G2 induced by
s |--> x * z^-1
t |--> y * z
u |--> x * y^-1 * z
> f2;
Homomorphism of GrpFP: G2 into GrpFP: G1 induced by
x |--> s^2 * u * t^-1
y |--> s^-1 * u^-1
z |--> s * u * t^-1
```
So the search for an isomorphism succeeded and returned the isomorphism and its inverse.
##### `Example: Isomorphism Search for Neumann's Groups (ex-8e44ee)`
Walter Neumann asked whether these two groups are isomorphic:
```magma
> G1 := FPGroup< x,y,z | x^2*y^5, x^14*z^23, (x^2,y), (x^2,z), x*y*z>;
> G2 := FPGroup;
```
It is a good idea to minimize the number of generators of $G1$ since there is clearly a redundant generator.
```magma
> G1s := Simplify(G1);
> Ngens(G1s);
2
> G1!G1s.1, G1!G1s.2;
x z
```
Now `SearchForIsomorphism` $(G1,\ G2,\ 15)$ will succeed, but will take many hours, and also use a lot of memory. In contrast to $G1$, it can sometimes help to introduce a new generator in the presentation for $G2$ if there are common substrings in relators.
```magma
> G2b := FPGroup< a,b,c | a*b^16*a*b^-7, a^4*b^7*a^-1*b^7, c=b^7>;
> isiso, f1, f2 := SearchForIsomorphism(G1s,G2b,4);
> isiso;
true
> f1;
Homomorphism of GrpFP: G1s into GrpFP: G2b induced by
G1s.1 |--> a * c^-1
G1s.2 |--> c
> f2;
Homomorphism of GrpFP: G2b into GrpFP: G1s induced by
a |--> G1s.1 * G1s.2
b |--> G1s.1^6 * G1s.2^10
c |--> G1s.2
```
### Quotient Groups
A powerful technique for investigating an fp-group is to study some of its quotients. For example, the action of a group $G$ on the cosets of a subgroup $H$ (where the index of $H$ in $G$ is finite) as described in the section on subgroups is an epimorphism of $G$ onto a permutation group of degree $[G:H]$. In this section further methods for constructing various types of quotient group are described. Specifically, machinery is provided for constructing abelian quotients, $p$-quotients, nilpotent quotients, soluble quotients, and $PSL(2, q)/PGL(2, q)$ quotients.
#### Abelian Quotient
Computing the quotient of an fp-group $G$ by its derived group is usually the first step in analyzing an fp-group. The ability to do this to finite-index subgroups of $G$ is a key step when trying to prove that a group is infinite. The intrinsics in this section compute information about the abelian quotient of an fp-group $G$.
**Intrinsics:**
`AbelianQuotient` $(G)$ : The maximal abelian quotient $G/G^\prime$ of the fp-group $G$ is returned as a group in category `GrpAb` (cf. chapter [Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)). The natural epimorphism $\pi:G\rightarrow G/G^\prime$ is returned as second value.
`HasFiniteAbelianQuotient` $(G)$ : Return `true` if the maximal abelian quotient of the fp-group $G$ is finite.
`ElementaryAbelianQuotient` $(G,~p)$ : The maximal $p$-elementary abelian quotient $Q$ is returned as a group $G$ in category `GrpAb` (cf. chapter [Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)). The natural epimorphism $\pi:G\rightarrow Q$ is returned as second value.
`AbelianQuotientInvariants` $(G)$ : Let $G$ be an fp-group or a subgroup of finite index of an fp-group. The elementary divisors of the derived quotient group $G/G'$ are returned as a sequence of integers.
`AbelianQuotientInvariants` $(G,~n)$ : Let $G$ be an fp-group, $n$ a small integer, and $N$ the normal subgroup of $G$ generated by the derived group $G'$ together with all $n$-th powers of elements of $G$. The elementary divisors of the quotient group $G/N$, are returned as a sequence of integers.
`AbelianQuotientPrimes` $(G)$ : Return a sequence of primes containing all prime divisors in the order of $G/G'$, if this is finite. Note that there may be primes in the return sequence which do not divide this order. If `[ 0 ]` is returned then $G/G'$ is infinite. If `[]` is returned, then $G$ is perfect.
`HasComputableAbelianQuotient` $(G)$ : Given an fp-group $G$, this intrinsic tests whether the abelian quotient of $G$ can be computed. If so, it returns the value `true`, the abelian quotient $A$ of $G$, and the natural epimorphism $\pi:G\ar A$. If the abelian quotient of $G$ cannot be computed, the value `false` is returned.
`HasInfiniteComputableAbelianQuotient` $(G)$ : Given an fp-group $G$, this function tests whether the abelian quotient of $G$ can be computed and is infinite. If so, it returns the value `true`, the abelian quotient $A$ of $G$, and the natural epimorphism $\pi:G\ar A$. If the abelian quotient of $G$ cannot be computed or if it is finite, the value `false` is returned.
**Notes:**
**(i)**
A number of variations on the above intrinsics may be found in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
**(ii)**
The above intrinsics require either a presentation for $G$, or the coset table for $G$ inside some supergroup of $G$ which has a presentation. If such a coset table cannot be computed, an error will result.
**(iii)**
Some functions may require the computation of a coset table. Experienced users can control the behaviour of an implicit coset enumeration by setting global parameters which can be changed using the function `SetGlobalTCParameters`.
**(iv)**
Intrinsics with names of the form `AbelianQuotientName` can be abbreviated to `AQName`, where name is `Quotient`, `Primes`, or `Invariants`.
**(v)**
Considerable effort has been put into computing the Smith Normal Form for large integer matrices that can arise when executing these intrinsics. An important application where such large matrices arise is when the input is the integer matrix resulting from the abelianisation of the presentation of a subgroup $H$ which has been constructed by rewriting the presentation of its parent group. If the index is large (even 50 or 100) then the presentation of $H$ may be huge.
**(vi)**
The intrinsic `HasInfiniteComputableAbelianQuotient` first checks the modular abelian invariants for a set of small primes. If for one of these primes the modular abelian quotient is trivial, then $A$ must be finite and the function returns without actually computing the abelian quotient. This heuristic may save time if only infinite quotients are of interest.
##### `Example: Structure of the Fibonacci Group F(7) (ex-12a8db)`
The Fibonacci group $F(7)$ has the following $2$-generator presentation:
$$
.
$$
The goal is to determine the structure of this group.
```magma
> F := FreeGroup(2);
> F7 := quo< F | a^2*b^-2*a^-1*b^-2*(a^-1*b^-1)^2,
> a*b*a*b^2*a*b*a*b^-1*(a*b^2)^2 >;
> F7;
Finitely presented group F7 on 2 generators
Relations
a^2 * b^-2 * a^-1 * b^-2 * a^-1 * b^-1 * a^-1 * b^-1 =Id(F7)
a * b * a * b^2 * a * b * a * b^-1 * a * b^2 * a * b^2 = Id()7
```
The first step is to determine the structure of the maximal abelian quotient of $F(7)$.
```magma
> AbelianQuotientInvariants(F7);
[ 29 ]
```
The maximal abelian quotient of $F(7)$ is cyclic of order 29. At this point there is no obvious way to proceed, so an attempt is made to determine the index of some subgroups.
```magma
> Index( F7, sub< F7 | a > );
1
```
This demonstrates that $F(7)$ is generated by $a$ and so must be cyclic. This fact coupled with the knowledge that its abelian quotient has order 29 proves that the group is cyclic of order 29.
##### `Example: Abelian Quotient Invariants of a Subgroup (ex-26e129)`
The group $G = (8, 7\ |\ 2, 3)$ is defined by the presentation
$$
< a, b \ | \ a^8, b^7, (ab)^2, (ab^{-1})^3 >.
$$
The goal here is to determine the abelian quotient invariants of the (rather famous) subgroup $H$ of $G$, generated by the words $a^2$ and $ab^{-1}$:
```magma
> G := FPGroup;
> H := sub< G | a^2, a * b^-1 >;
```
The fastest way to determine the order of the maximal 2-elementary abelian quotient of $H$ is to use the modular version of `AbelianQuotientInvariants`:
```magma
> #AbelianQuotientInvariants(H, 2);
1
```
So the maximal 2-elementary abelian quotient of $H$ has order $2^1$.
#### $p$-Quotient
Let $F$ be a finitely presented group, $p$ a prime, and $c$ a positive integer. A $p$-quotient algorithm constructs a consistent power-conjugate presentation for the largest $p$-quotient of $F$ having lower exponent-$p$ class at most $c$. The $p$-quotient algorithm used by Magma is part of the ANU $p$-Quotient program. For details of the algorithm, see [[NO96](https://docs.magma-maths.org/references.md#cite-newman-obrien)]. The result is returned as a group of type `GrpPC` (cf. chapter [Finite Soluble Groups](https://docs.magma-maths.org/FiniteGroups/FiniteSolubleGroups/index-finite-soluble-groups.md#chapgrppc)).
**Intrinsics:**
`pQuotient` $(G,\ p,\ c)$ : Given an fp-group $G$, a prime $p$, and a positive integer $c$, construct a pc-presentation for the largest $p$-quotient $Q$ of $G$ having lower exponent-$p$ class at most $c$. If $c$ is given as $0$ then the limit 127 is placed on the class. The prime $p$ must be less than $65536$. The intrinsic also returns the natural homomorphism $\pi$ from $G$ to $Q$, a sequence $S$ describing the definitions of the pc-generators of $Q$ and a flag indicating whether $Q$ is the maximal $p$-quotient of $G$.
**Notes:**
**(i)**
If the fp-group is a finite $p$-group then the $p$-quotient function can be used to construct a pc-presentation for the group.
**(ii)**
If the parameter `Metabelian` is set to `true` then a consistent pc-presentation is constructed for the largest metabelian $p$-quotient of $G$ having lower exponent-$p$ class at most $c$.
**(iii)**
Assume that the $p$-quotient $Q$ has order $p^n$, Frattini rank $d$, and that its generators are $a_1, \ldots, a_n$. Then the power-conjugate presentation constructed has the following additional structure. The set $\{a_1,\ldots,a_d\}$ is a generating set for $G$. For each $a_k$ in $\{a_{d+1},\ldots,a_n\}$, there is at least one relation whose right hand side is $a_k$. One of these relations is taken as the *definition* of $a_k$.
**(iv)**
The intrinsic has the following parameters in addition to `Metabelian`:
**-**
`Exponent` enables an exponent law, $x^m = 1$, to be imposed on the quotient being computed. It is set to a positive integer value $m$.
**-**
`Print` enables the user to obtain verbose output during the running of the intrinsic. Legal values are the integers 1, 2, and 3.
**-**
`Workspace` allows the user to specify the size of workspace that will be assigned to the $p$-quotient job.
**(v)**
A process version of the $p$-quotient algorithm is provided. This enables the user to create a restartable computation which gives the user complete control over its execution. For a description see chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
**(vi)**
The $k$-th term of the return value sequence $S$ is a sequence of two integers, describing the definition of the $k$-th pc-generator $Q.k$ of $Q$ as follows:-
**-**
If $S[k] = [0,r]$, then $Q.k$ is defined via the image of $G.r$ under $\pi$.
**-**
If $S[k] = [r,0]$, then $Q.k$ is defined via the power relation for $Q.r$.
**-**
If $S[k] = [r,s]$, then $Q.k$ is defined via the conjugate relation involving ${(Q.r)}^{Q.s}$.
##### `Example: Largest 2-Quotient of Class 6 (ex-81ff16)`
In this example the largest $2$-quotient of class $6$ for a two-generator, two-relator group is constructed.
```magma
> F := FreeGroup(2);
> G := quo< F | (b, a, a) = 1, (a * b * a)^4 = 1 >;
> Q, fQ := pQuotient(G, 2, 6);
> Order(Q);
524288
> fQ;
Mapping from: GrpFP: G to GrpPC: Q
```
##### `Example: 3-Quotient of a Group of Exponent 9 (ex-94855c)`
The largest $3$-quotient of class $6$ for a two-generator group of exponent $9$ will be constructed.
```magma
> F := FreeGroup(2);
> G := quo< F | a^3 = b^3 = 1 >;
> q := pQuotient(G, 3, 6: Print := 1, Exponent := 9);
Lower exponent-3 central series for G
Group: G to lower exponent-3 central class 1 has order 3^2
Group: G to lower exponent-3 central class 2 has order 3^3
Group: G to lower exponent-3 central class 3 has order 3^5
Group: G to lower exponent-3 central class 4 has order 3^7
Group: G to lower exponent-3 central class 5 has order 3^9
Group: G to lower exponent-3 central class 6 has order 3^11
```
##### `Example: Metabelian 5-Quotient (ex-e48428)`
The parameter `Metabelian`: will be used to construct a metabelian $5$-quotient of the group
$$
< a, b\ |\ a^{625} = b^{625} = 1, (b, a, b) = 1, (b, a, a, a, a) = (b, a)^5>.
$$
```magma
> F := FreeGroup(2);
> G := quo< F | a^625 = b^625 = 1, (b, a, b) = 1,
> (b, a, a, a, a) = (b, a)^5 >;
> q := pQuotient(G, 5, 20: Print := 1, Metabelian := true);
Lower exponent-5 central series for G
Group: G to lower exponent-5 central class 1 has order 5^2
Group: G to lower exponent-5 central class 2 has order 5^5
Group: G to lower exponent-5 central class 3 has order 5^8
Group: G to lower exponent-5 central class 4 has order 5^11
Group: G to lower exponent-5 central class 5 has order 5^12
Group: G to lower exponent-5 central class 6 has order 5^13
Group: G to lower exponent-5 central class 7 has order 5^14
Group: G to lower exponent-5 central class 8 has order 5^15
Group: G to lower exponent-5 central class 9 has order 5^16
Group: G to lower exponent-5 central class 10 has order 5^17
Group: G to lower exponent-5 central class 11 has order 5^18
Group: G to lower exponent-5 central class 12 has order 5^19
Group: G to lower exponent-5 central class 13 has order 5^20
Group completed. Lower exponent-5 central class = 13, order = 5^20
```
#### Nilpotent Quotient
A nilpotent quotient algorithm constructs, from a finite presentation of a group, a polycyclic presentation for a nilpotent quotient of the finitely presented group. The nilpotent quotient algorithm used by Magma is the ANU Nilpotent Quotient program, as described in [[Nic96](https://docs.magma-maths.org/references.md#cite-nickel-nq)]. The version included in Magma is Version 2.2 of January 2007.
The lower central series $G_0, G_1, \dots$ of a group $G$ can be defined inductively as $G_0 = G$, $G_i = [G_(i-1),G]$. The group $G$ is said to have nilpotency class $c$ if $c$ is the smallest non-zero integer such that $G_c = 1$. If $N$ is a normal subgroup of $G$ and $G/N$ is nilpotent, then $N$ contains $G_i$ for some non-negative integer $i$. $G$ has infinite nilpotent quotients if and only if $G/G_1$ (the maximal abelian quotient of $G$) is infinite and a prime $p$ divides a finite factor of a nilpotent quotient if and only if $p$ divides a cyclic factor of $G/G_1$.
**Intrinsics:**
`NilpotentQuotient` $(G,\ c)$ : The class $c$ nilpotent quotient of $G$ is computed as a group in category [`GrpGPC`](https://docs.magma-maths.org/FinitelyPresentedGroups/PolycyclicGroups/index-polycyclic-groups.md#chapgrpgpc), and is returned together with the epimorphism $\pi$ from $G$ onto this quotient. When $c$ is set to zero, the function attempts to compute the maximal nilpotent quotient of $G$.
**Notes:**
**(i)**
There are a number of parameters associated with this intrinsic and descriptions of these can be found in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
**(ii)**
The verbose identifier for this intrinsic is `NilpotentQuotient` which takes a small positive integer $n$ in the range $[0..3]$. Setting $n$ to be 0 turns printing off and this is the default value.
##### `Example: Class 2 Nilpotent Quotient (ex-0b5725)`
Here is a finitely presented group. The abelian quotient is infinite so the class 2 nilpotent quotient is computed.
```magma
> G := Group;
> AbelianQuotient(G);
Abelian Group isomorphic to Z/2 + Z/2 + Z
Defined on 3 generators
Relations:
2*$.1 = 0
2*$.2 = 0
> N := NilpotentQuotient(G,2); N;
GrpGPC : N of infinite order on 6 PC-generators
PC-Relations:
N.1^2 = N.3^2 * N.5,
N.2^2 = N.4 * N.6,
N.4^2 = Id(N),
N.5^2 = Id(N),
N.6^2 = Id(N),
N.2^N.1 = N.2 * N.4,
N.3^N.1 = N.3 * N.5,
N.3^N.2 = N.3 * N.6
```
##### `Example: Free Nilpotent Groups via Nilpotent Quotient (ex-470660)`
The free nilpotent group of rank $r$ and class $e$ is defined as $F / \gamma_{e+1}(F)$, where $F$ is a free group of rank $r$ and $\gamma_{e+1}(F)$ denotes the $(e+1)$ st term of the lower central series of $F$. The intrinsic `NilpotentQuotient` will be used to construct the free nilpotent group $N$ of rank 2 and class 3 as quotient of the free group $F$ of rank 2 and the natural epimorphism from $F$ onto $N$.
```magma
> F := FreeGroup(2);
> N<[x]>, pi := NilpotentQuotient(F, 3);
> N;
GrpGPC : N of infinite order on 5 PC-generators
PC-Relations:
x[2]^x[1] = x[2] * x[3],
x[2]^(x[1]^-1) = x[2] * x[3]^-1 * x[4],
x[3]^x[1] = x[3] * x[4],
x[3]^(x[1]^-1) = x[3] * x[4]^-1,
x[3]^x[2] = x[3] * x[5],
x[3]^(x[2]^-1) = x[3] * x[5]^-1
```
Next the nilpotency class of the quotient will be confirmed..
```magma
> NilpotencyClass(N);
3
```
##### `Example: Nilpotent Quotient of a Baumslag-Solitar Group (ex-f4b49e)`
The Baumslag–Solitar groups
$$
BS(p,q) = \langle\ a,\ b\ \vert\ ab^pa^{-1} = b^q \rangle\
$$
form a fascinating class of 1-relator groups. The nilpotent quotient of class 4 for $q = 4$, will be computed and the preimages of its generators identified.
```magma
> G := FPGroup;
> N,f := NilpotentQuotient(G,4);
> N;
GrpGPC : N of infinite order on 5 PC-generators
PC-Relations:
N.2^3 = N.3^2 * N.4^2 * N.5,
N.3^3 = N.4^2 * N.5^2,
N.4^3 = N.5^2,
N.5^3 = Id(N),
N.2^N.1 = N.2 * N.3,
N.2^(N.1^-1) = N.2 * N.3^2 * N.4^2 * N.5,
N.3^N.1 = N.3 * N.4,
N.3^(N.1^-1) = N.3 * N.4^2 * N.5^2,
N.4^N.1 = N.4 * N.5,
N.4^(N.1^-1) = N.4 * N.5^2
> for i := 1 to Ngens(N) do N.i @@ f; end for;
a
b
(b, a)
(b, a, a)
(b, a, a, a)
```
#### Soluble Quotient
A soluble quotient algorithm computes a consistent power-conjugate presentation (pc-presentation) of the largest finite soluble quotient of an fp-group, subject to certain algorithmic and user supplied restrictions. In this section a description of a basic version of the algorithm available within Magma in given. For more information the user is referred to chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
**Intrinsics:**
`SolvableQuotient/SolubleQuotient` $(G)$ : Given an fp-group $G$, the largest finite soluble quotient $Q$ is returned. The epimorphism $\phi : G \rightarrow Q$ is also returned.
`SolvableQuotient/SolubleQuotient` $(G, n)$ : Given an fp-group $G$ and a positive integer $n$, a soluble quotient $Q$ of order $n$ is returned. The $\phi : G \rightarrow Q$ is also returned. When the order of $Q$ is known in advance, this version can avoid work in proving that the final quotient is maximal.
`SolvableQuotient/SolubleQuotient` $(G, P)$ : Given an fp-group $G$ and a sequence $P$ containing primes this intrinsic calculates the largest soluble quotient whose order has prime divisors only in $P$. The epimorphism $\phi : G \rightarrow Q$ is also returned. This version is included since finding the possible primes at each step can be very expensive.
**Notes:**
**(i)**
A user wishing to get the most out of `SolubleQuotient` should note that there is a complex interface to it and the user should read the full description of this intrinsic, its variations, and the parameters available, in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
**(ii)**
In the case of `SolubleQuotient` $(G,~n)$, if $n>0$ is not the order of the maximal finite soluble quotient, then it may happen that no group of order $n$ can be found, since an epimorphic image of size $n$ may not be exhibited by the chosen series.
##### `Example: Soluble Quotient of a Two-Generator Group (ex-211a7e)`
The soluble quotient of the group
$$
\langle a,b \,\,|\,\, a^2, b^4,
ab^{-1}ab(abab^{-1})^5ab^2ab^{-2} \rangle
$$
will be computed.
```magma
> G := FPGroup< a, b | a^2, b^4,
> a*b^-1*a*b*(a*b*a*b^-1)^5*a*b^2*a*b^-2 >;
> Q := SolubleQuotient(G);
> Q;
GrpPC : Q of order 1920 = 2^7 * 3 * 5
PC-Relations:
Q.1^2 = Q.4,
Q.2^2 = Id(Q),
Q.3^2 = Q.6,
Q.4^2 = Id(Q),
Q.5^2 = Q.7,
Q.6^2 = Id(Q),
Q.7^2 = Id(Q),
Q.8^3 = Id(Q),
Q.9^5 = Id(Q),
Q.2^Q.1 = Q.2 * Q.3,
Q.3^Q.1 = Q.3 * Q.5,
Q.3^Q.2 = Q.3 * Q.6,
Q.4^Q.2 = Q.4 * Q.5 * Q.6 * Q.7,
Q.4^Q.3 = Q.4 * Q.6 * Q.7,
Q.5^Q.1 = Q.5 * Q.6,
Q.5^Q.2 = Q.5 * Q.7,
Q.5^Q.4 = Q.5 * Q.7,
Q.6^Q.1 = Q.6 * Q.7,
Q.8^Q.1 = Q.8^2,
Q.8^Q.2 = Q.8^2,
Q.9^Q.1 = Q.9^3,
Q.9^Q.2 = Q.9^4,
Q.9^Q.4 = Q.9^4
```
##### `Example: Order of a Large Soluble Quotient (ex-820eb8)`
Consider the group $G$ defined by the presentation
$$
\langle x, y \;\;|\;\; x^3, y^8, [x,y^4], x^{-1}yx^{-1}y^{-1}xyxy^{-1},
(xy^{-2})^2(x^{-1}y^{-2})^2(xy^2)^2(x^{-1}y^2)^2,
(x^{-1}y^{-2})^6(x^{-1}y^2)^6 \rangle.
$$
```magma
> G := FPGroup< x, y | x^3, y^8, (x,y^4), x^-1*y*x^-1*y^-1*x*y*x*y^-1,
> (x*y^-2)^2*(x^-1*y^-2)^2*(x*y^2)^2*(x^-1*y^2)^2, (x^-1*y^-2)^6*(x^-1*y^2)^6 >;
```
The soluble quotient algorithm will be applied to $G$ and its order computed.
```magma
> time Q := SolubleQuotient(G);
Time: 4.620
> Order(Q);
165888
```
Note that $165888 = 2^{11} \cdot 3^4$. Knowing the primes in advance makes the computation faster:
```magma
> time Q := SolubleQuotient(G, {2, 3});
Time: 1.900
```
Assuming that $G$ is finite the Todd–Coxeter procedure will used to find its order:
```magma
> Order(G);
165888
```
Hence the group $G$ is finite, soluble, and isomorphic to $Q$. Since $Q$ is defined by a pc-presentation anything can be computed for it and so it can be used to obtain information about $G$.
```magma
> #ConjugacyClasses(Q);
272
```
So $G$ has 272 conjugacy classes of elements.
#### Simple Group Quotients
The previous four methods for finding epimorphisms of an fp-group $G$ only apply when $G$ has a non-trivial abelian quotient, that is, when $G$ is not perfect. This and the next subsection present methods for the case in which $G$ is perfect. The first method searches through a list of non-abelian simple groups to find simple quotients of $G$ as permutation groups. It is similar to the algorithm used to find homomorphisms of an fp-group onto a permutation group. Currently (2019) the list of target simple groups consists of all non-abelian simple groups having order less than $10^{10}$. Such a list is dominated by ${\operatorname{PSL}}(2,q)$’s with $q$ odd. In the current implementation these ${\operatorname{PSL}}(2,q)$’s are treated as an infinite family rather than stored individually, and so continue beyond the above limit.
**Intrinsics:**
`SimpleQuotients(G, deg1, deg2, ord1, ord2)` :
`SimpleQuotients(G, ord1, ord2)` :
The homomorphism algorithm is used to find epimorphisms from $G$ onto non-abelian simple groups in a fixed list. The arguments $deg1$ and $deg2$ are respectively lower and upper bounds for the degree of the image group. If the degree arguments are not present then bounds of $5$ and $10^{10}$ are used. The arguments $ord1$ and $ord2$ are, respectively, lower and upper bounds for the orders of the image group. (Setting $ord2$ low enough is particularly important if a quick search is required. If $ord1$ is not given then it defaults to $1$. The return value is a list of sequences containing the epimorphisms found. Each sequence contains epimorphisms onto one simple group.
The parameter `Limit` limits the number of successful searches to be performed by `Homomorphisms`. The default value is 1, so by default the search terminates with the first simple group found to be a homomorphic image of $G$.
**Notes:**
**(i)**
The parameter `Limit` limits the number of successful searches that are to be performed by `Homomorphisms`. The default value is 1, so by default the search terminates with the first simple group found to be a homomorphic image of $G$.
**(ii)**
The parameter `HomLimit` limits the number of homomorphisms that will be searched for by any particular call to `Homomorphisms`. It defaults to zero, so that all homomorphisms for any group found will be returned in that case.
**(iii)**
The parameter `Family` selects sublists of the main list to search. Possible values of this parameter are `"All"`, `"PSL"`, `"PSL2"`, `"Mathieu"`, `"Alt"`, `"PSp"`, `"PSU"`, `"Other"`, and `"notPSL2"`; sets of these strings are also allowed, which searches on the union of the appropriate sublists.
**(iii)**
A process version of `SimpleQuotients` creates a restartable version of the computation so that it returns a result to the user each time a new epimorphism is found. It involves three intrinsics:
- `SimpleQuotientProcess` has the same arguments as `SimpleQuotient`. It initialises the process and returns a name $P$ for future reference to the process.
- `NextSimpleQuotient` $(P)$, where $P$ is the process name continues the search until the next epimorphism is found whereupon it is returned and the process is suspended until the next call.
- `IsEmptySimpleQuotientProcess` $(P)$, where $P$ is the process name, returns `false` when the search has been completed and `true` otherwise.
Full details can be found in chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp).
##### `Example: Simple Quotients of a Perfect Group (ex-af2186)`
The following finitely presented group is perfect so a search is made for simple quotients. The first search is for simple quotients of order not exceeding $10^5$ and degree not exceeding 100.
```magma
> G := FPGroup;
> IsPerfect(G);
true
> L := SimpleQuotients(G, 1, 100, 2, 10^5 : Limit := 2);
> #L;
2
> for x in L do CompositionFactors(Image(x[1])); end for;
G
| A(1, 13) = L(2, 13)
1
G
| A(2, 3) = L(3, 3)
1
> L[2,1];
Homomorphism of GrpFP: F into GrpPerm: $, Degree 13, Order
2^4 * 3^3 * 13 induced by
F.1 |--> (1, 10, 4, 5, 11, 8, 3, 6, 7, 12, 9, 13, 2)
F.2 |--> (2, 10, 4)(3, 6, 7)(5, 11, 13)(8, 12, 9)
F.3 |--> (1, 10)(2, 5)(3, 12)(8, 13)
> #L[2];
2
```
The simple groups L(2,13) and L(3,3) are quotients, with two inequivalent homomorphisms onto the second group. The process mechanism will be used to search for projective unitary groups of order up to $10^6$ and having permutation representations of degree not greater then 100.
```magma
> P := SimpleQuotientProcess(G, 1, 100, 2, 10^6 : Family:="PSU");
> IsEmptySimpleQuotientProcess(P);
false
> eps, info := SimpleEpimorphisms(P);
> info;
<65, 62400, PSU(3, 4)>
```
The group PSU(3,4) of order 62400 and degree 65 is a quotient of $G$. The process is restarted.
```magma
> NextSimpleQuotient(~P);
> IsEmptySimpleQuotientProcess(P);
true
```
There are no more unitary group quotients within the given degree and order limits.
#### ${\rm L}_2$-Quotients
Given an fp-group $G$, the $L_2$-quotient algorithm of Plesken, Fabianska, and Jambor computes all quotients of $G$ which are isomorphic to the $2$-dimensional linear groups $PSL(2, q)$ or $PGL(2, q)$ simultaneously for any prime power $q$. The algorithm can handle the case of infinitely many quotients, and also works for very large prime powers. In practice it is fast when the number of generators is small and can be used to show that many fp-groups are infinite. While the simple groups $PSL(2, q)$ represent just one dimension in one family out of 16 families of simple groups, in a given range of orders, they are very numerous compared to other simple groups. See [[Jam12](https://docs.magma-maths.org/references.md#cite-jamborphd)] or [[Jam15](https://docs.magma-maths.org/references.md#cite-jambor-l2quot)] for more information about the $L_2$-quotient algorithm.
In this chapter only part of the $L_2$-quotient machinery is discussed while computing $L_3$- and $U_3$-quotients is not discussed at all. The reader is referred to chapter [Finitely Presented Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/index-fpgroups.md#chapgrpfp) for a complete treatment of the $L_2/L_3/U_3$-quotient machinery.
The term $L_2$-quotient will include both of the groups $PSL(2, q)$ and $PGL(2, q)$. $L_2$-quotients have their own Magma type: `L2Quotient`. Some fp-groups $G$ have infinitely many $L_2$-quotients and this is indicated by one of the $L_2$-quotient names `L_2(infty^k), L_2(p^(infty^d))`, or `L_2(infty^(infty^d))`.
**Intrinsics:**
`L2Quotients` $(G)$ : Given a fp group $G$, a sequence containing all ${\rm L}_2$-quotients is returned.
`GetMatrices` $(Q)$ : For a finite $L_2$-quotient $Q$ of $G$, (that is, a quotient $L_2(p^k)$ or ${\operatorname{PGL}}(2,p^k)$), this intrinsic returns a matrix group $H$ and a sequence $A$ of $2\times 2$ matrices in $H$, where $A[i]$ corresponds to the $i$-th generator of $G$. Note that $G \rightarrow H$, $G.i \rightarrow A[i]$ does not in general define a homomorphism, but the induced map $G \rightarrow H/Z(H)$ does.
**Notes:**
**(i)**
At the moment the algorithm does not return images onto the groups $PSL(2, 2)$, $PSL(2, 3)$, and $PSL(2, 4) = PSL(2 , 5)$.
**(ii)**
Infinite quotients should be interpreted as follows:
- **Quotients of type** ${\rm L}_2(\infty^k)$ : If $G$ has a quotient ${\rm L}_2(\infty^k)$, then for almost all (all but finitely many) primes $p$, $G$ has finitely many quotients of type ${{\operatorname{PSL}}}(2,p^r)$ or ${{\operatorname{PGL}}}(2,p^{r/2})$ with $r \leq k$. So ${\rm L}_2(\infty^k)$ is a mnemonic, where $\infty$ in the base stands for infinitely many primes, and $k$ stands for the highest possible exponent.
- **Quotients of type** ${\rm L}_2(p^{\infty^d})$ : If $G$ has a quotient ${\rm L}_2(p^{\infty^d})$, then there are infinitely many positive integers $k$ such that $G$ has a quotient of type ${{\operatorname{PSL}}}(2,p^k)$ or ${{\operatorname{PGL}}}(2,p^k)$. So ${\rm L}_2(p^{\infty^d})$ is a mnemonic, where $\infty$ in the exponent stands for infinitely many possible exponents. The parameter $d$ describes the degree of infinity, and is omitted if $d = 1$.
- **Quotients of type** ${\rm L}_2(\infty^{\infty^d})$ : If $G$ has a quotient ${\rm L}_2(\infty^{\infty^d})$, then for almost all primes $p$ and infinitely many positive integers $k$, $G$ has a quotient of type ${{\operatorname{PSL}}}(2,p^k)$ or ${{\operatorname{PGL}}}(2,p^k)$. So ${\rm L}_2(\infty^{\infty^d})$ is a mnemonic, where $\infty$ in the base stands for infinitely many primes, and $\infty$ in the exponent stands for infinitely many possible exponents. The parameter $d$ describes the degree of infinity, and is omitted if $d = 1$. These quotients can be further investigated using the methods `AddGroupRelations`, `AddRingRelations`, and `SpecifyCharacteristic`.
##### `Example: L2-Quotients of FP-Groups (ex-5313fb)`
The $L_2$-quotients of a certain finitely-presented group will be computed.
```magma
> G := FPGroup< a,b | a^2, b^3, (a*b)^7, (a,b)^11 >;
> L2Quotients(G);
[
L_2(43)
]
> H := FPGroup< a,b,c | a^3, b^7, c^19, (a*b)^2, (a*c)^2, (b*c)^2, (a*b*c)^2 >;
> L2Quotients(H);
[
L_2(113)
]
```
This means that $G$ has ${\operatorname{PSL}}(2,43)$ as quotient, but no other ${\operatorname{PSL}}(2,q)$ or ${\operatorname{PGL}}(2,q)$ is a quotient of $G$. Similarly, the only $L_2$-quotient of $H$ is ${\operatorname{PSL}}(2,113)$.
The next example has more quotients:
```magma
> G := FPGroup< a,b | a^2, b^3, (a*b)^16, (a,b)^11 >;
> L2Quotients(G);
[
PGL(2,23),
PGL(2,23),
PGL(2,463)
]
```
Here ${\operatorname{PGL}}(2,23)$ occurs twice. This means that there are two epimorphisms of $G$ onto ${\operatorname{PGL}}(2,23)$ which do not differ by an automorphism of ${\operatorname{PGL}}(2,23)$. In other words, the kernels of the epimorphisms are distinct.
##### `Example: Groups with Infinitely Many L2-Quotients (ex-0eaade)`
Some groups have infinitely many $L_2$-quotients. This is indicated by one of the $L_2$-quotients `L_2(infty^k), L_2(p^(infty^d))`, or `L_2(infty^(infty^d))`, as the follow examples illustrate.
```magma
> G := FPGroup< a,b,c | a^3, b^7, (a*b)^2, (a*c)^2, (b*c)^2, (a*b*c)^2 >;
> L2Quotients(G);
[
L_2(infty^6)
]
> H := FPGroup;
> L2Quotients(H);
[
L_2(3^infty)
]
> K := FPGroup< a,b | a^3*b^3 >;
> L2Quotients(K);
[
L_2(infty^infty)
]
```
##### `Example: Matrix Groups of an L2-Quotient (ex-ba85d0)`
The following example illustrates how the quotient matrix groups can be accessed.
```magma
> H := FPGroup< a,b,c | a^3, b^7, c^19, (a*b)^2, (a*c)^2, (b*c)^2, (a*b*c)^2 >;
> quot := L2Quotients(H); quot;
[
L_2(113)
]
> H, A := GetMatrices(quot[1]);
> H;
MatrixGroup(2, GF(113))
Generators:
[ 0 1]
[112 112]
[ 0 85]
[109 24]
[102 104]
[ 63 72]
> A;
[
[ 0 1]
[112 112],
[ 0 85]
[109 24],
[102 104]
[ 63 72]
]
```
## Finitely Presented Groups
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/introduction.md)
- [Overview of Facilities](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/introduction.md#overview-of-facilities)
- [Construction of an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md)
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#introduction)
- [Quotient Group Constructor](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#quotient-group-constructor)
- [`quo< F | R >: GrpFP, List → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)
- [`G / H: GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#operation-op-div-grpfp-grpfp)
- [Example: Sym(4) as a Quotient of a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-94ed98)
- [Example: Quotient Constructor Without a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-803aa2)
- [Example: Quotient of a Triangle Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-8457bf)
- [The FP-Group Constructor](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#the-fp-group-constructor)
- [`FPGroup< X | R >: List(Var), List(GrpFPRel) → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-1-confpgroup)
- [Example: Binary Tetrahedral Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-388271)
- [Example: Group Generated by Three Involutions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-cb4daf)
- [Example: Coxeter's Parametrised Families of Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-4c068c)
- [Accessing the Defining Generators and Relations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#accessing-the-defining-generators-and-relations)
- [`G . i: GrpFP, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#operation-operation-grpfp-rngintelt-grpfpelt)
- [`Generators(G): GrpFP → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-generators-grpfp)
- [`NumberOfGenerators(G): GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-numberofgenerators-grpfp)
- [`Ngens(G): GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-ngens-grpfp)
- [`PresentationLength(G): GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-presentationlength-grpfp)
- [`Relations(G): GrpFP → [ GrpFPRel ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-relations-grpfp)
- [Operations on Words](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#operations-on-words)
- [`Eliminate(u, x, v): GrpFPElt, GrpFPElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-eliminate-grpfpelt-grpfpelt-grpfpelt)
- [`Eliminate(U, x, v): { GrpFPElt }, GrpFPElt, GrpFPElt → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-eliminate-grpfpelt-grpfpelt-grpfpelt-2)
- [`Match(u, v, f): GrpFPElt, GrpFPElt, RngIntElt → BoolElt, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-match-grpfpelt-grpfpelt-rngintelt)
- [`RotateWord(u, n): GrpFPElt, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-rotateword-grpfpelt-rngintelt)
- [`Substitute(u, f, n, v): GrpFPElt, RngIntElt, RngIntElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-substitute-grpfpelt-rngintelt-rngintelt-grpfpelt)
- [`Subword(u, f, n): GrpFPElt, RngIntElt, RngIntElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-subword-grpfpelt-rngintelt-rngintelt)
- [Example: Operations on Words in a Free Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-1c9bc2)
- [Operations on Presentations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#operations-on-presentations)
- [`AddGenerator(G): GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-addgenerator-grpfp)
- [`AddGenerator(G, w): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-addgenerator-grpfp-grpfpelt)
- [`AddRelation(G, r): GrpFP, RelElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-addrelation-grpfp-relelt)
- [`AddRelation(G, g): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-addrelation-grpfp-grpfpelt)
- [`AddRelation(G, r, i): GrpFP, RelElt, RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-addrelation-grpfp-relelt-rngintelt)
- [`AddRelation(G, g, i): GrpFP, GrpFPElt, RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-addrelation-grpfp-grpfpelt-rngintelt)
- [`DeleteGenerator(G, x): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-deletegenerator-grpfp-grpfpelt)
- [`DeleteRelation(G, r): GrpFP, RelElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-deleterelation-grpfp-relelt)
- [`DeleteRelation(G, g): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-deleterelation-grpfp-grpfpelt)
- [`DeleteRelation(G, i): GrpFP, RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-deleterelation-grpfp-rngintelt)
- [`ReplaceRelation(G, s, r): GrpFP, RelElt, RelElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-replacerelation-grpfp-relelt-relelt)
- [`ReplaceRelation(G, h, r): GrpFP, GrpFPElt, RelElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-replacerelation-grpfp-grpfpelt-relelt)
- [`ReplaceRelation(G, s, g): GrpFP, RelElt, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-replacerelation-grpfp-relelt-grpfpelt)
- [`ReplaceRelation(G, h, g): GrpFP, GrpFPElt, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-replacerelation-grpfp-grpfpelt-grpfpelt)
- [`ReplaceRelation(G, i, r): GrpFP, RngIntElt, RelElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-grpfp-replacerelation)
- [`ReplaceRelation(G, i, g): GrpFP, RngIntElt, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-replacerelation-grpfp-rngintelt-grpfpelt)
- [Example: Replacing a Relation](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#example-ex-c80efd)
- [Simplification](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md)
- [Automatic Simplification](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#automatic-simplification)
- [`ReduceGenerators(G): GrpFP → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-reducegenerators)
- [`Simplify(G: parameters): GrpFP → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify)
- [Example: Simplifying the Fibonacci Group F(8)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#example-ex-6d0638)
- [`SimplifyLength(G: parameters): GrpFP → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplifylength)
- [Interactive Simplification](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#interactive-simplification)
- [`TietzeProcess(G: parameters): GrpFP → Process(Tietze)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-tietzeprocess-grpfp)
- [`ShowOptions(~P : parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-showoptions-grpfptietzeproc-ref)
- [`SetOptions(~P : parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-setoptions-grpfptietzeproc-ref)
- [`Simplify(~P : parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-simplify-grpfptietzeproc-ref)
- [`SimplifyPresentation(~P : parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-simplifypresentation-grpfptietzeproc-ref)
- [`SimplifyLength(~P : parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-simplifylength-grpfptietzeproc-ref)
- [`Eliminate(~P: parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-eliminate-grpfptietzeproc-ref)
- [`EliminateGenerators(~P: parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-eliminategenerators-grpfptietzeproc-ref)
- [`Search(~P: parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-search-grpfptietzeproc-ref)
- [`SearchEqual(~P: parameters): GrpFPTietzeProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#procedure-searchequal-grpfptietzeproc-ref)
- [`Group(P): GrpFPTietzeProc → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-tietze-group)
- [`NumberOfGenerators(P): GrpFPTietzeProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-numberofgenerators-grpfptietzeproc)
- [`Ngens(P): GrpFPTietzeProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-ngens-grpfptietzeproc)
- [`NumberOfRelations(P): GrpFPTietzeProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-numberofrelations-grpfptietzeproc)
- [`Nrels(P): GrpFPTietzeProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-nrels-grpfptietzeproc)
- [`PresentationLength(P): GrpFPTietzeProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-presentationlength-grpfptietzeproc)
- [Example: Interactive Simplification of the Fibonacci Group F(7)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#example-ex-1eeac7)
- [Example: Reducing a Generating Set](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#example-ex-df6881)
- [Standard Constructions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md)
- [Familiar Groups as FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#familiar-groups-as-fp-groups)
- [`AbelianFPGroup([n₁,...,nᵣ]): [ RngIntElt ] → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-abelianfpgroup-rngintelt)
- [`AlternatingFPGroup(n): RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-alternatingfpgroup-rngintelt)
- [`Alt(GrpFP, n): Cat, RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-alt-cat-rngintelt)
- [`BraidFPGroup(n): RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-braidfpgroup-rngintelt)
- [`CoxeterFPGroup(t): MonStgElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-coxeterfpgroup-monstgelt)
- [`CyclicFPGroup(n): RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-cyclicfpgroup-rngintelt)
- [`DihedralFPGroup(n): RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-dihedralfpgroup-rngintelt)
- [`ExtraSpecialFPGroup(p, n : parameters): RngIntElt, RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-extraspecialfpgroup-rngintelt-rngintelt)
- [`SymmetricFPGroup(n): RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-symmetricfpgroup-rngintelt)
- [`Sym(GrpFP, n): Cat, RngIntElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-sym-cat-rngintelt)
- [Example: Standard Groups as FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-936a87)
- [Construction of Extensions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#construction-of-extensions)
- [`Darstellungsgruppe(G): GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-darstellungsgruppe-grpfp)
- [`DirectProduct(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-directproduct-grpfp-grpfp)
- [`DirectProduct(Q): [ GrpFP ] → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-directproduct-grpfp)
- [`FreeProduct(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-freeproduct-grpfp-grpfp)
- [`FreeProduct(Q): [ GrpFP ] → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-freeproduct-grpfp)
- [Example: Maximal Central Extension of a Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-dc16bb)
- [Example: Direct Product of FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-bdaee4)
- [Conversion to FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#conversion-to-fp-groups)
- [`FPGroup(G): GrpPerm → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-fpgroup)
- [`FPGroup(G): GrpMat → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-fpgroup-grpmat)
- [Example: FP-Group from a Permutation Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-185d54)
- [`FPGroupStrong(G): GrpPerm → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-fpgroupstrong)
- [`FPGroupStrong(G): GrpMat → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-fpgroupstrong-grpmat)
- [`FPGroupStrong(G, N): GrpPerm, GrpPerm → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-fpgroupstrong-grpperm-grpperm)
- [Example: FP-Group on Strong Generators](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-f1a62b)
- [`FPGroup(G): GrpPC → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-fpgroup-grppc)
- [`FPGroup(G): GrpGPC → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-fpgroup-grpgpc)
- [`FPGroup(G): GrpAb → GrpFP, Hom(Grp)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-fpgroup-grpab)
- [Example: Converting a Polycyclic Group to an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-a91a69)
- [`CoxeterFPGroup(W): GrpFPCox → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-coxeterfpgroup-conv)
- [`CoxeterFPGroup(W): GrpPermCox → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-coxeterfpgroup-conv-2)
- [Example: Coxeter Group as an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#example-ex-1b08d9)
- [Properties of an FP-Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md)
- [Cardinality](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#cardinality)
- [`Order(G: parameters): GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-grpfp-1-order)
- [`FactoredOrder(G: parameters): GrpFP → [ ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-grpfp-1-factoredorder)
- [Example: Order of a Finitely Presented Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#example-ex-134948)
- [`IsInfiniteFPGroup(G : parameters): GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-isinfinitefpgroup-grpfp)
- [Example: Proving a Group Infinite](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#example-ex-fea44f)
- [`HasPositiveH1Dimension(G, phi : parameters): GrpFP, HomGrp → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-haspositiveh1dimension-grpfp-homgrp)
- [Small Cancellation Conditions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#small-cancellation-conditions)
- [`SmallCancellationConditions(G): GrpFP → RngIntElt, RnIntElt, FldRatElt)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-smallcancellationconditions-grpfp)
- [Example: Small Cancellation Conditions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#example-ex-2f799a)
- [Largeness](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#largeness)
- [`IsLarge(G, L, U:parameters): GrpFP, RngIntElt, RngIntElt → BoolElt, GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-grpfp-1-islarge)
- [Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md)
- [Specification of a Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#specification-of-a-subgroup)
- [`sub< G | L >: GrpFP, List → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructor-grpfp-1-sub)
- [`sub< G | f >: GrpFP, Hom(Grp) → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructor-constructor-sub-grpfp-hom-grp-grpfp)
- [`ncl< G | L >: GrpFP, List → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructor-grpfp-1-ncl)
- [`ncl: GrpFP, Hom(Grp) → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructor-constructor-ncl-grpfp-hom-grp-grpfp)
- [`CommutatorSubgroup(G): GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-commutatorsubgroup-grpfp)
- [`DerivedSubgroup(G): GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-derivedsubgroup-grpfp)
- [`DerivedGroup(G): GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-derivedgroup-grpfp)
- [Example: Subgroup of Index 448](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-6af151)
- [Example: Subgroup Defined by a Homomorphism](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-4fd961)
- [The Todd-Coxeter Algorithm](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#the-todd-coxeter-algorithm)
- [`ToddCoxeter(G, H: parameters): GrpFP, GrpFP → RngIntElt, Map, RngIntElt, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-1-toddcoxeter)
- [`Index(G, H: parameters): GrpFP, GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-1-index)
- [`FactoredIndex(G, H: parameters): GrpFP, GrpFP → [ ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-factoredindex-grpfp-grpfp)
- [Example: Todd-Coxeter Enumeration of 448 Cosets](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-0e7ed5)
- [Example: Todd-Coxeter for the Harada-Norton Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-c75263)
- [Example: Orders of a Parametrised Family of Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-4e7bab)
- [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#interactive-coset-enumeration)
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#introduction)
- [Constructing and Modifying a Coset Enumeration Process](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructing-and-modifying-a-coset-enumeration-process)
- [`CosetEnumerationProcess(G, H: parameters): GrpFP, GrpFP → GrpFPCosetEnumProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess)
- [`AddRelator(~P, w): GrpFPCosetEnumProc, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-addrelator)
- [`AddSubgroupGenerator(~P, w): GrpFPCosetEnumProc, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-addsubgroupgenerator)
- [`SetProcessParameters(~P: parameters): GrpFPCosetEnumProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setprocessparameters)
- [Starting and Restarting an Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#starting-and-restarting-an-enumeration)
- [`StartEnumeration(~P: parameters): GrpFPCosetEnumProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-startenumeration)
- [`RedoEnumeration(~P: parameters): GrpFPCosetEnumProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-redoenumeration)
- [`CanRedoEnumeration(P): GrpFPCosetEnumProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-canredoenumeration)
- [`ContinueEnumeration(~P: parameters): GrpFPCosetEnumProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-continueenumeration)
- [`CanContinueEnumeration(P): GrpFPCosetEnumProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cancontinueenumeration)
- [`ResumeEnumeration(~P: parameters): GrpFPCosetEnumProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-resumeenumeration)
- [Accessing Information](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#accessing-information)
- [`CosetsSatisfying(P : parameters): GrpFPCosetEnumProc → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying)
- [`CosetSatisfying(P : parameters): GrpFPCosetEnumProc → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetsatisfying)
- [`CosetTable(P): GrpFPCosetEnumProc → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosettable)
- [`HasValidCosetTable(P): GrpFPCosetEnumProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable)
- [`HasClosedCosetTable(P): GrpFPCosetEnumProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasclosedcosettable)
- [`HasCompleteCosetTable(P): GrpFPCosetEnumProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-hascompletecosettable-grpfpcosetenumproc)
- [`ExcludedConjugate(P): GrpFPCosetEnumProc → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-excludedconjugate-grpfpcosetenumproc)
- [`ExcludedConjugates(P): GrpFPCosetEnumProc → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-excludedconjugates-grpfpcosetenumproc)
- [`ExistsCosetSatisfying(P : parameters): GrpFPCosetEnumProc → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-existscosetsatisfying-grpfpcosetenumproc)
- [`ExistsExcludedConjugate(P): GrpFPCosetEnumProc → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-existsexcludedconjugate-grpfpcosetenumproc)
- [`ExistsNormalisingCoset(P): GrpFPCosetEnumProc → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-existsnormalisingcoset-grpfpcosetenumproc)
- [`ExistsNormalizingCoset(P): GrpFPCosetEnumProc → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-existsnormalizingcoset-grpfpcosetenumproc)
- [`Group(P): GrpFPCosetEnumProc → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-group-grpfpcosetenumproc)
- [`Index(P): GrpFPCosetEnumProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-index)
- [`HasValidIndex(P): GrpFPCosetEnumProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidindex)
- [`MaximalNumberOfCosets(P): GrpFPCosetEnumProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-maximalnumberofcosets-grpfpcosetenumproc)
- [`Subgroup(P): GrpFPCosetEnumProc → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-subgroup-grpfpcosetenumproc)
- [`TotalNumberOfCosets(P): GrpFPCosetEnumProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-totalnumberofcosets-grpfpcosetenumproc)
- [Example: Interactive Coset Table for Harada-Norton](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-bfc054)
- [Example: Copying a Coset Enumeration Process](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-e1cc15)
- [Example: Normaliser Element from a Partial Coset Table](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-ef1fe7)
- [Example: Excluded Conjugates in a Coset Table](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-0f03e7)
- [Implicit Invocation of the Todd-Coxeter Algorithm](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#implicit-invocation-of-the-todd-coxeter-algorithm)
- [`SetGlobalTCParameters(: parameters)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters)
- [`UnsetGlobalTCParameters()`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-unsetglobaltcparameters)
- [Example: Implicit Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-de08eb)
- [Coset Spaces and Tables](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md)
- [Coset Tables](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#coset-tables)
- [`CosetTable(G, H: parameters): GrpFP, GrpFP → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosettable-grpfp-grpfp)
- [`CosetTableToRepresentation(G, T): GrpFP, Map → Map, GrpPerm, Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-cosettabletorepresentation)
- [`CosetTableToPermutationGroup(G, T): GrpFP, Map → GrpPerm`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosettabletopermutationgroup-grpfp-map)
- [Example: Coset Table in the Infinite Dihedral Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-ae13a4)
- [Coset Spaces: Induced Homomorphism](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#coset-spaces-induced-homomorphism)
- [`CosetAction(G, H): Grp, Grp → Hom(Grp), GrpPerm, Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetaction-grp-grp)
- [`CosetAction(V): GrpFPCos → Hom(Grp), GrpPerm`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetaction-grpfpcos)
- [`CosetImage(G, H): Grp, Grp → GrpPerm`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetimage-grp-grp)
- [`CosetImage(V): GrpFPCos → GrpPerm`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetimage-grpfpcos)
- [`CosetKernel(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-cosetkernel)
- [`CosetKernel(V): GrpFPCos → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetkernel-grpfpcos)
- [Example: Induced Homomorphism for Co1](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-a27ede)
- [Example: Permutation Representation of G2(3)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-6d4ee4)
- [Coset Spaces: Construction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#coset-spaces-construction)
- [`CosetSpace(G, H: parameters): GrpFP, GrpFP → GrpFPCos`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetspace-grpfp-grpfp)
- [`RightCosetSpace(G, H: parameters): GrpFP, GrpFP → GrpFPCos`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-rightcosetspace-grpfp-grpfp)
- [`LeftCosetSpace(G, H: parameters): GrpFP, GrpFP → GrpFPCos`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-leftcosetspace-grpfp-grpfp)
- [Coset Spaces: Elementary Operations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#coset-spaces-elementary-operations)
- [`H * g: GrpFP, GrpFPElt → GrpFPCosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-times-grpfp-grpfpelt)
- [`C * g: GrpFPCosElt, GrpFPElt → GrpFPCosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-times-grpfpcoselt-grpfpelt)
- [`C * D: GrpFPCosElt, GrpFPCosElt → GrpFPCosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-times-grpfpcoselt-grpfpcoselt)
- [`g in C: GrpFPElt, GrpFPCosElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-in-grpfpelt-grpfpcoselt)
- [`g notin C: GrpFPElt, GrpFPCosElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-notin-grpfpelt-grpfpcoselt)
- [`C1 eq C2: GrpFPCosElt, GrpFPCosElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-eq-grpfpcoselt-grpfpcoselt)
- [`C1 ne C2: GrpFPCosElt, GrpFPCosElt → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-op-ne-grpfpcoselt-grpfpcoselt)
- [Accessing Information](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#accessing-information)
- [`# V: GrpFPCos → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#operation-operation-grpfpcos-rngintelt)
- [`Action(V): GrpFPCos → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-action-grpfpcos)
- [` @ T: GrpFPCosElt, GrpFPElt, Map → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#literal-literal-i-w-t-grpfpcoselt-grpfpelt-map-grpfpelt)
- [`T(i, w): Map, GrpFPCosElt, GrpFPElt → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#literal-t-map-grpfpcoselt-grpfpelt)
- [`ExplicitCoset(V, i): GrpFPCos, RngIntElt → GrpFPCosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-explicitcoset-grpfpcos-rngintelt)
- [`IndexedCoset(V, w): GrpFPCos, GrpFPElt → GrpFPCosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-indexedcoset-grpfpcos-grpfpelt)
- [`IndexedCoset(V, C): GrpFPCos, GrpFPCosElt → GrpFPCosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-indexedcoset-grpfpcos-grpfpcoselt)
- [`Group(V): GrpFPCos → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-group-grpfpcos)
- [`Subgroup(V): GrpFPCos → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-subgroup-grpfpcos)
- [`IsComplete(V): GrpFPCos → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-iscomplete-grpfpcos)
- [`ExcludedConjugates(V): GrpFPCos → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-excludedconjugates-grpfpcos)
- [`ExcludedConjugates(T): Map → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-excludedconjugates-map)
- [`Transversal(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-transversal-grpfp-grpfp)
- [`RightTransversal(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-righttransversal)
- [Example: Accessing a Coset Table](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-e36cb8)
- [Example: Working with a Coset Space](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-886b74)
- [Example: Derived Subgroup via Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-13317b)
- [Example: Adding Excluded Conjugates](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-8d212e)
- [Double Coset Spaces: Construction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#double-coset-spaces-construction)
- [`DoubleCoset(G, H, g, K ): GrpFP, GrpFP, GrpFPElt, GrpFP → GrpFPDcosElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-doublecoset-grpfp-grpfp-grpfpelt-grpfp)
- [`DoubleCosets(G, H, K): GrpFP, GrpFP, GrpFP → { GrpFPDcosElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-doublecosets)
- [Example: Double Cosets in the Infinite Dihedral Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-b7c1be)
- [Coset Spaces: Selection of Cosets](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#coset-spaces-selection-of-cosets)
- [`CosetsSatisfying(T, S: parameters): Map, { GrpFPElt } → { GrpFPCosElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-cosetssatisfying)
- [`CosetSatisfying(T, S: parameters): Map, { GrpFPElt } → { GrpFPCosElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-cosetsatisfying)
- [`CosetsSatisfying(V, S: parameters): GrpFPCos, { GrpFPElt } → { GrpFPCosElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetssatisfying-grpfpcos-grpfpelt)
- [`CosetSatisfying(V, S: parameters): GrpFPCos, { GrpFPElt } → { GrpFPCosElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-cosetsatisfying-grpfpcos-grpfpelt)
- [Example: Selecting Cosets by a Condition](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-c086e8)
- [Constructing a Presentation for a Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#constructing-a-presentation-for-a-subgroup)
- [`Rewrite(G, H : parameters): GrpFP, GrpFP → GrpFP, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-rewrite)
- [`Rewrite(G, ~H : parameters): GrpFP, GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#procedure-grpfp-1-rewrite-var)
- [Example: Presentation for an Index 3 Subgroup](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-e234d8)
- [Example: Presentation on New Generators](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-490fe5)
- [Subgroups of Finite Index](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md)
- [Low Index Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#low-index-subgroups)
- [`LowIndexSubgroups(G, R : parameters): GrpFP, RngIntElt → [ GrpFP ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-lowindexsubgroups)
- [`LowIndexSubgroups(G, R : parameters): GrpFP, RngIntElt → [ { GrpFPElt } ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-lowindexsubgroups-grpfp-rngintelt)
- [`LowIndexSubgroups(G, R: parameters): GrpFP, → [ GrpFP ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-lowindexsubgroups-grpfp-rngintelt-rngintelt)
- [`LowIndexSubgroups(G, R: parameters): GrpFP, → [ { GrpFPElt } ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-lowindexsubgroups-grpfp-rngintelt-rngintelt-2)
- [Example: Tutte's 8-Cage and the Conder Graph](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-2e2f0b)
- [Example: Conjugacy Classes via Low Index Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-18108c)
- [`LowIndexProcess(G, R : parameters): GrpFP, RngIntElt → Process(Lix)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-lowindexprocess)
- [`LowIndexProcess(G, R: parameters): GrpFP, < RngIntElt, RngIntElt > → Process(Lix)`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-lowindexprocess-grpfp-rngintelt-rngintelt)
- [`NextSubgroup(~P): GrpFPLixProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup)
- [`NextSubgroup(~P, ~G): GrpFPLixProc, GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-nextsubgroup-grpfplixproc-grpfp-ref)
- [`ExtractGroup(P): GrpFPLixProc → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-extractgroup-grpfplixproc)
- [`ExtractGenerators(P): GrpFPLixProc → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-extractgenerators-grpfplixproc)
- [`IsEmpty(P): GrpFPLixProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-isempty-grpfplixproc)
- [`IsValid(P): GrpFPLixProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-isvalid-lix)
- [Example: Low Index Subgroups of a Triangle Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-fd99df)
- [Example: PSL(2,8) as a Quotient of a Triangle Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-55eb8b)
- [Example: Proving a Group Is Infinite](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-dd899e)
- [`LowIndexNormalSubgroups(G, n: parameters): GrpFP, RngIntElt → [ Rec ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-lowindexnormalsubgroups-grpfp-rngintelt)
- [Operations for Subgroups of Finite Index](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operations-for-subgroups-of-finite-index)
- [`H ^ u: GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-op-pow-grpfp-grpfpelt)
- [`Conjugate(H, u): GrpFP, GrpFPElt → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-conjugate-grpfp-grpfpelt)
- [`H meet K: GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-meet)
- [`Core(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-core)
- [`GeneratingWords(G, H): GrpFP, GrpFP → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-generatingwords)
- [`MaximalOvergroup(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-maximalovergroup)
- [`MinimalOvergroup(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-minimalovergroup)
- [`H ^ G: GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-op-pow-grpfp-grpfp)
- [`NormalClosure(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-normalclosure)
- [`Normaliser(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-normaliser)
- [`Normalizer(G, H): GrpFP, GrpFP → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-normalizer-grpfp-grpfp)
- [`SchreierGenerators(G, H : parameters): GrpFP, GrpFP → { GrpFPElt }`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-schreiergenerators)
- [`SchreierSystem(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-schreiersystem)
- [`Transversal(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-transversal)
- [`Transversal(G, H, K): GrpFP, GrpFP, GrpFP → {@ GrpFPElt @}, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-transversal-3)
- [Example: Subgroup Constructions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-793825)
- [Example: Schreier Generators](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-f2320c)
- [Properties of Subgroups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#properties-of-subgroups)
- [`u in H: GrpFPElt, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-in)
- [`u notin H: GrpFPElt, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-notin)
- [`H eq K: GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-eq)
- [`H ne K: GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-ne)
- [`H subset K: GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-subset)
- [`IsSubgroup(H,K): GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-issubgroup-grpfp-grpfp)
- [`H notsubset K: GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-notsubset)
- [`IsConjugate(G, H, K): GrpFP, GrpFP, GrpFP → BoolElt, GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-isconjugate)
- [`IsNormal(G, H): GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-isnormal)
- [`IsMaximal(G, H): GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-ismaximal)
- [`IsSelfNormalizing(G, H): GrpFP, GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-isselfnormalizing)
- [Example: Subgroup Predicates in a Space Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-afcd46)
- [Example: Maximal Subgroups of the Hall-Janko Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-bf7b3f)
- [Finite FP-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md)
- [Concrete Representations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#concrete-representations)
- [`PermutationGroup(G): GrpFP → GrpPerm, GrpHom`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#function-permutationgroup-grpfp)
- [`PCGroup(G): GrpFP → GrpPC, GrpHom`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#function-pcgroup-grpfp)
- [The Cayley Graph](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#the-cayley-graph)
- [`CayleyGraph(G): GrpFP → GrphVertSet, GrphEdgeSeto`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#function-cayleygraph-grpfp)
- [`SchreierGraph(G, H): Grp, Grp → Grph, GrphVertSet, GrphEdgeSet`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#function-schreiergraph-grp-grp)
- [Example: Cayley Graph of a Dihedral Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/finite-fp-groups.md#example-ex-6ec7fa)
- [Homomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md)
- [General Remarks](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#general-remarks)
- [Construction of Homomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#construction-of-homomorphisms)
- [`hom< P -> G | S >: Struct, Struct → Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#constructor-constructor-hom-struct-struct-map)
- [`IsSatisfied(U, E): { RelElt }, [ GrpElt ] → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-issatisfied-relelt-grpelt)
- [`IsSatisfied(U, E): { GrpFPElt }, [ GrpElt ] → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-issatisfied-grpfpelt-grpelt)
- [`IsSatisfied(U, E): [ RelElt ], [ GrpElt ] → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-issatisfied-relelt-grpelt-2)
- [`IsSatisfied(U, E): [ GrpFPElt ], [ GrpElt ] → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-issatisfied)
- [Accessing Homomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#accessing-homomorphisms)
- [`w @ f: GrpFPElt, Map → GrpElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#literal-literal-w-f-grpfpelt-map-grpelt)
- [`f(w): Map, GrpFPElt → GrpElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#literal-f-map-grpfpelt)
- [`H @ f: GrpFP, Map → Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#literal-literal-h-f-grpfp-map-grp)
- [`f(H): Map, GrpFP → Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#literal-f-map-grpfp)
- [`g @@ f: GrpElt, Map → GrpFPElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#literal-literal-g-f-grpelt-map-grpfpelt)
- [`H @@ f: Grp, Map → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#literal-literal-h-f-grp-map-grpfp)
- [`Domain(f): Map → Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-domain-map)
- [`Codomain(f): Map → Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-codomain-map)
- [`Image(f): Map → Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-image-map)
- [`Kernel(f): Map → Grp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-kernel)
- [Example: Braid Group onto Sym(5)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-63663e)
- [Constructing Homomorphisms onto Finite Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#constructing-homomorphisms-onto-finite-groups)
- [`Homomorphisms(F, G, A : parameters): GrpFP, GrpPerm, GrpPerm → [ HomGrp ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-homomorphisms)
- [`Homomorphisms(F, G : parameters): GrpFP, GrpPerm → [ HomGrp ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-homomorphisms-grpfp-grpperm)
- [Example: Finding Homomorphisms onto Finite Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-731266)
- [`Homomorphisms(F, G, A : parameters): GrpFP, GrpPC, GrpPC → [ HomGrp ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-homomorphisms-grpfp-grppc-grppc)
- [`Homomorphisms(F, G : parameters): GrpFP, GrpPC → [ HomGrp ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-homomorphisms-grpfp-grppc)
- [`HomomorphismsProcess(F, G, A : parameters): GrpFP, GrpPerm, GrpPerm → GrpFPHomsProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-homomorphismsprocess)
- [`HomomorphismsProcess(F, G : parameters): GrpFP, GrpPerm → GrpFPHomsProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-homomorphismsprocess-grpfp-grpperm)
- [`NextElement(~P): GrpFPHomsProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#procedure-grpfp-1-nextelement)
- [`IsEmpty(P): GrpFPHomsProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isempty-hom)
- [`IsValid(P): GrpFPHomsProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isvalid-hom)
- [`DefinesHomomorphism(P): GrpFPHomsProc → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-defineshomomorphism)
- [`Homomorphism(P): GrpFPHomsProc → HomGrp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-homomorphism)
- [`# P: GrpFPHomsProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#operation-operation-grpfphomsproc-rngintelt)
- [Example: Braid Group onto PSL(2, 16)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-9b82f6)
- [Example: Proving Infiniteness Interactively](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-3b7c60)
- [Searching for Isomorphisms](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#searching-for-isomorphisms)
- [`SearchForIsomorphism(F, G, m : parameters): GrpFP, GrpFP, RngIntElt → BoolElt, HomGrp, HomGrp`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-searchforisomorphism)
- [Example: Isomorphism Search for Hillman's Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-d9442c)
- [Example: Isomorphism Search for Neumann's Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-4192a7)
- [Quotient Group Methods](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md)
- [Abelian Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#abelian-quotient)
- [`AbelianQuotient(G): GrpFP → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotient)
- [`ElementaryAbelianQuotient(G, p): GrpFP, RngIntElt → GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-elementaryabelianquotient)
- [`AbelianQuotientInvariants(G): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotientinvariants)
- [`AQInvariants(G): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqinvariants-grpfp)
- [`AbelianQuotientInvariants(H): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotientinvariants-sub)
- [`AQInvariants(H): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqinvariants-grpfp-2)
- [`AbelianQuotientInvariants(G, T): GrpFP, Map → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-abelianquotientinvariants-grpfp-map)
- [`AQInvariants(G, T): GrpFP, Map → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqinvariants-grpfp-map)
- [`AbelianQuotientInvariants(G, n): GrpFP, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotientinvariants-mod)
- [`AQInvariants(G, n): GrpFP, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqinvariants-grpfp-rngintelt)
- [`AbelianQuotientInvariants(H, n): GrpFP, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-abelianquotientinvariants-grpfp-rngintelt)
- [`AQInvariants(H, n): GrpFP, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqinvariants-grpfp-rngintelt-2)
- [`AbelianQuotientInvariants(G, T, n): GrpFP, Map, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-abelianquotientinvariants-grpfp-map-rngintelt)
- [`AQInvariants(G, T, n): GrpFP, Map, RngIntElt → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqinvariants-grpfp-map-rngintelt)
- [`HasComputableAbelianQuotient(G): GrpFP → BoolElt, GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-hascomputableabelianquotient)
- [`HasInfiniteComputableAbelianQuotient(G): GrpFP → BoolElt, GrpAb, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-hasinfinitecomputableabelianquotient-grpfp)
- [`IsPerfect(G): GrpFP → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-isperfect-grpfp)
- [`TorsionFreeRank(G): GrpFP → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-torsionfreerank-grpfp)
- [Example: Structure of the Fibonacci Group F(7)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-5b646e)
- [Example: Abelian Quotient Invariants Modulo 2](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-818c7a)
- [`HasFiniteAbelianQuotient(G): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-hasfiniteabelianquotient-grpfp)
- [`HasFiniteAQ(G): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-hasfiniteaq-grpfp)
- [`AQPrimes(G): GrpFP → [ RngIntElt ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-aqprimes-grpfp)
- [$p$-Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#p-quotient)
- [`pQuotient(F, p, c: parameters): GrpFP, RngIntElt, RngIntElt → GrpPC, Map, SeqEnum, BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-pquotient)
- [Example: Largest 2-Quotient of Class 6](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-e7285c)
- [Example: Largest 3-Quotient of Class 6](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-090d87)
- [Example: Metabelian 5-Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-bccae5)
- [Example: Largest 2-Generator Group of Exponent 5](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-a5910e)
- [`HaspQuotientDefinitions(G): GrpPC → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-haspquotientdefinitions)
- [`pQuotientDefinitions(G): GrpPC → SeqEnum`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-pquotientdefinitions-grppc)
- [$p$-Quotient Process](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#p-quotient-process)
- [`pQuotientProcess(F, p, c: parameters): GrpFP, RngIntElt, RngIntElt → Process`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-pquotientprocess)
- [`NextClass(~P : parameters): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-grpfp-nextclass)
- [`NextClass(~P, k : parameters): GrpPCpQuotientProc, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-nextclass-grppcpquotientproc-rngintelt-ref)
- [Using $p$-Quotient Interactively](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#using-p-quotient-interactively)
- [`StartNewClass(~P: parameters): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-startnewclass-grppcpquotientproc-ref)
- [`Tails(~P: parameters): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-tails-grppcpquotientproc-ref)
- [`Tails(~P, k: parameters): GrpPCpQuotientProc, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-tails-grppcpquotientproc-rngintelt-ref)
- [`Consistency(~P: parameters): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-consistency-grppcpquotientproc-ref)
- [`Consistency(~P, k: parameters): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-consistency-grppcpquotientproc-ref-2)
- [`CollectRelations(~P): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-collectrelations-grppcpquotientproc-ref)
- [`ExponentLaw(~P : parameters): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-exponentlaw-grppcpquotientproc-ref)
- [`ExponentLaw(~P, Start, Fin: parameters): GrpPCpQuotientProc, RngIntElt, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-exponentlaw-grppcpquotientproc-rngintelt-rngintelt-ref)
- [`EliminateRedundancy(~P): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-eliminateredundancy-grppcpquotientproc-ref)
- [`Display(P): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-display-grppcpquotientproc)
- [`Display(P, DisplayLevel): GrpPCpQuotientProc, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-display-grppcpquotientproc-rngintelt)
- [`RevertClass(~P): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-revertclass-grppcpquotientproc-ref)
- [`pCoveringGroup(~P): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-pcoveringgroup-grppcpquotientproc-ref)
- [`pCoveringGroup(G): GrpPC → GrpPC`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-pcoveringgroup-grppc)
- [`GeneratorStructure(P): GrpPCpQuotientProc`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-generatorstructure-grppcpquotientproc)
- [`GeneratorStructure(P, Start, Fin): GrpPCpQuotientProc, RngIntElt, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-generatorstructure-grppcpquotientproc-rngintelt-rngintelt)
- [`Jacobi(~P, c, b, a, ~r): GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt → RngIntElt ->`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-jacobi-grppcpquotientproc-rngintelt-rngintelt-rngintelt-ref)
- [`Jacobi(~P, c, b, a): GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt → RngIntElt ->`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-jacobi-grppcpquotientproc-rngintelt-rngintelt-rngintelt-ref-2)
- [`Collect(P, Q): GrpPCpQuotientProc, [ ] → [ RngIntElt ] ->`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-collect-grppcpquotientproc-rngintelt-rngintelt)
- [`EcheloniseWord(~P, ~r): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-echeloniseword-grppcpquotientproc-ref)
- [`EcheloniseWord(~P): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-echeloniseword-grppcpquotientproc-ref-2)
- [`SetDisplayLevel(~P, Level): GrpPCpQuotientProc, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-setdisplaylevel-grppcpquotientproc-rngintelt-ref)
- [`ExtractGroup(P): GrpPCpQuotientProc → GrpPC`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-extractgroup-grppcpquotientproc)
- [`Order(P): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-order-grppcpquotientproc)
- [`FactoredOrder(P): GrpPCpQuotientProc → [ ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-factoredorder-grppcpquotientproc)
- [`NumberOfPCGenerators(P): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-numberofpcgenerators-grppcpquotientproc)
- [`pClass(P): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-pclass-grppcpquotientproc)
- [`NuclearRank(G): GrpPC → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-nuclearrank-grppc)
- [`NuclearRank(P): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-nuclearrank-grppcpquotientproc)
- [`pMultiplicatorRank(G): GrpPC → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-pmultiplicatorrank-grppc)
- [`pMultiplicatorRank(P): GrpPCpQuotientProc → RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-pmultiplicatorrank-grppcpquotientproc)
- [Example: Extending a 3-Quotient Interactively](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-ea9994)
- [Example: Bounding Definitions of Pcp Generators](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-0cb2c6)
- [Example: Class 7 Quotient of R(2, 5)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-371e40)
- [Example: Interactively Extending a Class 6 Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-0ff68f)
- [Nilpotent Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#nilpotent-quotient)
- [`NilpotentQuotient(G, c: parameters): GrpFP, RngIntElt → GrpGPC, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-nilpotentquotient)
- [Example: Class 2 Nilpotent Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-7067cc)
- [Example: Free Nilpotent Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-8a4940)
- [Example: Nilpotent Quotients of Baumslag-Solitar Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-b77879)
- [`SetVerbose("NilpotentQuotient", n): MonStgElt, RngIntElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-setverbose-monstgelt-rngintelt-nilpotentquotient)
- [Example: Maximal Nilpotent Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-85800d)
- [Soluble Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#soluble-quotient)
- [`SolvableQuotient(G : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solvablequotient-grpfp-rngintelt)
- [`SolubleQuotient(G : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solublequotient-grpfp-rngintelt)
- [Example: Soluble Quotient of a Two-Generator Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-a7ae66)
- [`SolvableQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solvablequotient-grpfp-rngintelt-2)
- [`SolubleQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solublequotient-grpfp-rngintelt-2)
- [`SolvableQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solvablequotient-grpfp-set)
- [`SolubleQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solublequotient-grpfp-set)
- [Example: Order of a Soluble Quotient](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-130896)
- [Soluble Quotient Advanced](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#soluble-quotient-advanced)
- [Introduction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#introduction)
- [Construction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#construction)
- [Calculating the Relevant Primes](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#calculating-the-relevant-primes)
- [The Intrinsics](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#the-intrinsics)
- [`SolubleQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solublequotient-grpfp-rngintelt-3)
- [`SolvableQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solvablequotient-grpfp-rngintelt-3)
- [`SolubleQuotient(F : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solublequotient-grpfp-rngintelt-4)
- [`SolvableQuotient(F : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solvablequotient-grpfp-rngintelt-4)
- [`SolubleQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solublequotient-grpfp-set-2)
- [`SolvableQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-solvablequotient-grpfp-set-2)
- [Simple Group Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#simple-group-quotients)
- [`SimpleQuotients(F, deg1, deg2, ord1, ord2: parameters): GrpFP, RngIntElt, RngIntElt, RngIntElt, RngIntElt → List`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-simplequotients)
- [`SimpleQuotients(F, ord1, ord2: parameters): GrpFP, RngIntElt, RngIntElt → List`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-simplequotients-grpfp-rngintelt-rngintelt)
- [`SimpleQuotients(F, ord2: parameters): GrpFP, RngIntElt → List`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-simplequotients-grpfp-rngintelt)
- [`SimpleQuotientProcess(F, deg1, deg2, ord1, ord2: parameters): GrpFP, RngIntElt, RngIntElt, RngIntElt, RngIntElt → Rec`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-simplequotientprocess)
- [`NextSimpleQuotient(~P): Rec`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#procedure-grpfp-1-nextsimplequotient)
- [`IsEmptySimpleQuotientProcess(P): Rec → BoolElt`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-isemptysimplequotientprocess)
- [`SimpleEpimorphisms(P): Rec → SeqEnum, Tup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-simpleepimorphisms)
- [Example: Simple Quotients of a Perfect Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-b2e434)
- [The ${\rm L}_2$-Quotient Algorithm](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#the-rm-l-2-quotient-algorithm)
- [Basic Usage](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#basic-usage)
- [`L2Quotients(G): GrpFP → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-l2quotients-grpfp)
- [Example: Computing L2-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-656deb)
- [`GetMatrices(Q): L2Quotient → GrpMat, SeqEnum`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-getmatrices-l2quotient)
- [Example: Matrix Groups of L2-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-d6f5a7)
- [Example: Finite Quotients of the Modular Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-0ea7a6)
- [Example: L2-Quotients of Coxeter Presentations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-3ecfb1)
- [Intermediate Usage](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#intermediate-usage)
- [`L2Quotients(M): AlgMatElt → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-l2quotients-algmatelt)
- [Example: L2-Quotients from a Coxeter Matrix](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-cc44da)
- [`L2Quotients(G): GrpFP → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-l2quotients-grpfp-2)
- [Example: L2-Quotients with Exact Orders](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-39db62)
- [Advanced Usage](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#advanced-usage)
- [`L2Quotients(G): GrpFP → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-l2quotients-grpfp-3)
- [Handling Infinite ${\rm L}_2$-quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#handling-infinite-rm-l-2-quotients)
- [`SpecifyCharacteristic(Q, n): L2Quotient, RngIntElt → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-specifycharacteristic-l2quotient-rngintelt)
- [Example: Specifying the Characteristic](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-dd2847)
- [`AddGroupRelations(Q, R): L2Quotient, [ GrpFPElt ] → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-addgrouprelations-l2quotient-grpfpelt)
- [Example: Adding Group Relations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-6e7481)
- [Example: Adding Group Relations in Characteristic 3](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-d05ba8)
- [`AddRingRelations(Q, R): L2Quotient, [ RngMPolElt ] → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-addringrelations-l2quotient-rngmpolelt)
- [Example: Adding Ring Relations](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-6b7dfd)
- [Infinite L2 Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#infinite-l2-quotients)
- [`HasInfinitePSL2Quotient(G): GrpFP → BoolElt, SeqEnum`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-hasinfinitepsl2quotient-grpfp)
- [Example: Infinite and Finite L2-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-5b2c63)
- [The ${\rm L}_3{\rm U}_3$-Quotient Algorithm](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#the-rm-l-3-rm-u-3-quotient-algorithm)
- [`L3Quotients(G: parameters): GrpFP → [ L3Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-l3quotients-grpfp)
- [`SpecifyCharacteristic(Q, p): L2Quotient, RngIntElt → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-specifycharacteristic-l2quotient-rngintelt-2)
- [`SpecifyCharacteristic(Q, p): L3Quotient, RngIntElt → [ L3Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-specifycharacteristic-l3quotient-rngintelt)
- [`AddGroupRelations(Q, r): L2Quotient, [ GrpFPElt ] → [ L2Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-addgrouprelations-l2quotient-grpfpelt-2)
- [`AddGroupRelations(Q, r): L3Quotient, [ GrpFPElt ] → [ L3Quotient ]`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-addgrouprelations-l3quotient-grpfpelt)
- [`GetMatrices(Q): L2Quotient → GrpMat`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-getmatrices-l2quotient-2)
- [`GetMatrices(Q): L3Quotient → GrpMat`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-getmatrices-l3quotient)
- [Example: Computing L3- and U3-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-0214ac)
- [Example: Infinitely Many L3-Quotients](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#example-ex-ba1556)
- [$KG$-Modules](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md)
- [`GModulePrimes(G, A): GrpFP, GrpFP → SetMulti`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#function-grpfp-1-gmoduleprimes)
- [`GModulePrimes(G, A, B): GrpFP, GrpFP, GrpFP → SetMulti`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#function-grpfp-1-gmoduleprimes-3)
- [`GModule(G, A, p): GrpFP, GrpFP, RngIntElt → ModGrp, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#function-grpfp-1-gmodule)
- [`GModule(G, A, B, p): GrpFP, GrpFP, GrpFP, RngIntElt → ModGrp, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#function-grpfp-1-gmodule-3)
- [`GModule(G, A, B): GrpFP, GrpFP, GrpFP → ModGrp, Map`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#function-gmodule-grpfp-grpfp-grpfp)
- [`Pullback(f, N): Map, ModGrp → GrpFP`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#function-grpfp-1-pullback)
- [Example: KG-Modules of a Finitely Presented Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#example-ex-00eade)
- [Example: Relevant Primes for Module Construction](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#example-ex-c74200)
- [Some Developed Examples](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/developed-examples.md)
- [Example: Finiteness of the Fibonacci Group F(9)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/developed-examples.md#example-ex-c85753)
- [Example: Identifying PSL(3, 7):2 from a Presentation](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/developed-examples.md#example-ex-82d19f)
- [Bibliography](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/bibliography.md)
### Introduction
This Chapter documents the functions designed for computing with finitely-presented groups (fp-groups for short). The name of the corresponding Magma category is `GrpFP`. The intrinsics considered here are designed for computations in the area that is sometimes referred to as *combinatorial group theory*.
#### Overview of Facilities
The facilities provided for fp-groups fall into a number of natural groupings:
- The construction of fp-groups in terms of generators and relations
- The simplification of presentations using Tietze transformations
- The Todd-Coxeter procedure for constructing the coset table of a subgroup. This is the basis for many fp-group algorithms
- A collection of tools for computing with subgroups having (modest) finite index in a group, where the subgroups are represented by coset tables
- The construction of all subgroups having index less than some (small) specified bound
- The Reidemeister-Schreier algorithm for constructing presentations of subgroups
- The determination of various properties of fp-groups such as *perfect*, *finite/infinite*, *small cancellation*, and *large*
- A search for homomorphisms of an fp-group onto a permutation group or a pc-group
- A test for the isomorphism of two fp-groups
- The construction of particular types of quotient groups: abelian quotient, $p$-quotient, nilpotent quotient, soluble quotient, simple group quotient, and $L_2$/$L_3$/$U_3$ quotients
- The construction of representations of an fp-group corresponding to actions on coset spaces and elementary abelian sections
This chapter contains detailed information about the intrinsics provided for general fp-groups. The much shorter chapter [Introduction to Fp-Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/IntroductionToFPGroups/index-introduction-to-fpgroups.md#chapgrpfpint), “Introduction of Finitely Presented Groups”, gives a basic description of the more important intrinsics for fp-groups and should serve to get users started. That chapter typically does not list the parameters of an intrinsic and omits the more specialised intrinsics which are mainly useful when generic intrinsics fail.
Free groups are treated in Chapter [Free Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/index-free-groups.md#chapgrpfree) and it should be read in conjunction with this chapter. Automatic groups and hyperbolic groups are discussed in Chapter [Automatic and Hyperbolic Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AutomaticAndHyperbolicGroups/index-automatic-and-hyperbolic-groups.md#chapgrpatc).
For a description of the fundamental algorithms for finitely presented groups, the reader should consult Sims [[Sim94](https://docs.magma-maths.org/references.md#cite-sims-fpgroups)] or Holt [[HEO05](https://docs.magma-maths.org/references.md#cite-holt-handbook)].
### Construction of an FP-Group
An fp-group is normally constructed either by giving a presentation in terms of generators and relations or it is defined to be a subgroup or quotient group of an existing fp-group. However, fp-groups may be also created from finite permutation and matrix groups. Finally, Magma has separate types for a number of families of fp-groups that satisfy a condition such as being polycyclic. Again functions are provided to convert a member of one of these families into a general fp-group.
#### Introduction
The construction of fp-groups utilises the fact that every group is a quotient of some free group. Thus, two general fp-group constructors are provided: [`FreeGroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FreeGroups/creation.md#function-grpfp-1-freegroup)`(n)` which constructs a free group of rank $n$, and [`quo< F | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)`< F | R >` which constructs the quotient of group $F$ by the normal subgroup defined by the relations $R$.
The naming of generators presents special difficulties since they are not always used in a consistent manner in the mathematical literature. A generator name is used in two distinct ways. Firstly, it plays the role of a *variable* having as its value a designated generator of $G$. Secondly, it appears as the *symbol* designating the specified generator whenever elements of the group are output. These two uses are separated in the Magma semantics. In Magma, a standard indexing notation is provided for referencing the generators of any fp-group $G$. Thus, `G.i` denotes the $i$-th generator of $G$. However, users may give individual names to the generators by means of the *generator*-assignment. Suppose that the group $G$ is defined on $r$ generators. Then if the right hand side of the following statement creates a group, the special assignment
```
> G< v_1, ..., v_r> := \construction;
```
is equivalent to the statements
```
> G := \construction;
> v_1 := G.1;
> ...
> v_r := G.r;
```
It should be noted that when the fp-group $G$ is created as the quotient of the group $F$, any names that the user may have associated with the generators of $F$ will not be associated with the corresponding generators of $G$. If this were allowed, then it would violate the fundamental principle that every object is viewed as belonging to a *unique* structure.
#### Quotient Group Constructor
A group with non-trivial relations is constructed as a quotient of an existing group, usually a free group. For convenience, the necessary free group may be constructed in-line.
##### `quo< F | R >: GrpFP, List -> GrpFP, Hom(Grp)`
Given an fp-group $F$, and a set of relations $R$ in the generators of $F$, construct the quotient $G$ of $F$ by the normal subgroup of $F$ defined by $R$. The group $G$ is defined by means of a presentation which consists of the relations for $F$ (if any), together with the additional relations defined by the list $R$. The expression defining $F$ may be either simply the name of a previously constructed group, or an expression defining an fp-group. If $R$ is a list then each term of the list is either a *word*, a *relation*, a *relation list*, or a *subgroup* of $F$.
A *word* is interpreted as a relator.
A *relation* consists of a pair of words, separated by ‘=’. (See above.)
A *relation list* consists of a list of words, where each pair of adjacent words is separated by ‘=’: $w_1 = w_2 = \cdots = w_r$. This is interpreted as the set of relations $w_1 = w_r, \ldots, w_{r - 1} = w_r$. Note that the relation list construct is only meaningful in the context of the `quo`-constructor.
A *subgroup* $H$ appearing in the list $R$ contributes its generators to the relation set for $G$, i.e. each generator of $H$ is interpreted as a relator for $G$. The group $F$ may be referred to by the special symbol \$ in any word appearing to the right of the ‘`|`’ symbol in the `quo`-constructor. Also, in the context of the `quo`-constructor, the identity element may be represented by the digit `1`.
The function returns:
**(a)**
The quotient group $G$;
**(b)**
The natural homomorphism $\phi : F \rightarrow G$.
This function may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters).
##### `G / H: GrpFP, GrpFP -> GrpFP`
Given a subgroup $H$ of the group $G$, construct the quotient of $G$ by the normal closure $N$ of $H$. The quotient is formed by taking the presentation for $G$ and including the generating words of $H$ as additional relators.
##### `Example: Sym(4) as a Quotient of a Free Group (ex-94ed98)`
The symmetric group of degree 4 may be represented as a two generator group with presentation $< a, b \ | \ a^2, b^3, (ab)^4 >$. Giving the relations as a list of relators, the presentation would be specified as:
```magma
> F := FreeGroup(2);
> G, phi := quo< F | a^2, b^3, (a*b)^4 >;
```
Alternatively, giving the relations as a relations list, the presentation would be specified as:
```magma
> F := FreeGroup(2);
> G, phi := quo< F | a^2 = b^3 = (a*b)^4 = 1>;
```
Finally, giving the relations in the form of a set of relations, this presentation would be specified as:
```magma
> F := FreeGroup(2);
> rels := { a^2, b^3, (a*b)^4 };
> G, phi := quo< F | rels >;
```
##### `Example: Quotient Constructor Without a Free Group (ex-803aa2)`
A group may be defined using the [`quo< F | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)-constructor without first assigning a free group. The \$ symbol is used to reference the group whose quotient is being formed.
```magma
> S4 := quo< FreeGroup(2) | $.1^2, $.2^3, ($.1*$.2)^4 >;
> S4;
Finitely presented group S4 on 2 generators
Relations
x^2 = Id(S4)
y^3 = Id(S4)
(x * y)^4 = Id(S4)
```
##### `Example: Quotient of a Triangle Group (ex-8457bf)`
We illustrate the use of the [`quo< F | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)-constructor in defining the quotient of a group other than a free group.
```magma
> F := FPGroup< x, y | x^2 = y^3 = (x*y)^7 = 1 >;
> F;
Finitely presented group F on 2 generators
Relations
x^2 = Id(F)
y^3 = Id(F)
(x * y)^7 = Id(F)
> G := quo< F | (x, y)^8 >;
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^3 = Id(G)
(a * b)^7 = Id(G)
(a, b)^8 = Id(G)
> Order(G);
10752
```
#### The FP-Group Constructor
For convenience, a constructor is provided which allows the user to define an fp-group in a single step.
##### `FPGroup< X | R >: List(Var), List(GrpFPRel) -> GrpFP, Hom(Grp)`
Given a list $X$ of variables $x_1, \ldots, x_r$, and a list of relations $R$ over these generators, first construct the free group $F$ on the generators $x_1, \cdots, x_r$ and then construct the quotient of $F$ corresponding to the normal subgroup of $F$ defined by the relations $R$. The syntax for the relations $R$ is the same as for the [`quo< F | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)-constructor. The function returns:
**(a)**
The quotient group $G$;
**(b)**
The natural homomorphism $\phi : F \rightarrow G$.
##### `Example: Binary Tetrahedral Group (ex-388271)`
We illustrate the [`FPGroup< X | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-1-confpgroup)-constructor by defining the binary tetrahedral group in terms of the presentation $< r, s \ | \ r^3 = s^3 = (rs)^2 >:$
```magma
> G := FPGroup< r, s | r^3 = s^3 = (r*s)^2 >;
```
##### `Example: Group Generated by Three Involutions (ex-cb4daf)`
Again, using the [`FPGroup< X | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-1-confpgroup)-constructor, the group $< r, s, t \ | \ r^2, s^2, t^2, rst = str = trs >$ would be specified as:
```magma
> G := FPGroup;
```
##### `Example: Coxeter's Parametrised Families of Groups (ex-4c068c)`
In our final example we illustrate the use of functions to represent parametrised families of groups. In the notation of Coxeter, the symbol $(l, m | n, k)$ denotes the family of groups having presentation
$$
< a, b\ |\ a^l, b^m, (a*b)^n, (a*b^{-1})^k >.
$$
```magma
> Glmnk := func< l, m, n, k | FPGroup< a, b | a^l, b^m, (a*b)^n, (a*b^-1)^k > >;
> G := Glmnk(3, 3, 4, 4);
> G;
Finitely presented group G on 2 generators
Relations
a^3 = Id(G)
b^3 = Id(G)
(a * b)^4 = Id(G)
(a * b^-1)^4 = Id(G)
> Order(G);
168
> G := Glmnk(2, 3, 4, 5);
> G;
Finitely presented group G on 2 generators
Relations
a^2 = Id(G)
b^3 = Id(G)
(a * b)^4 = Id(G)
(a * b^-1)^5 = Id(G)
> Order(G);
1
```
Thus $(2, 3 \;\;|\;\; 4, 5 )$ is the trivial group.
#### Accessing the Defining Generators and Relations
The functions in this group provide access to basic information stored for a finitely-presented group $G$.
##### `G . i: GrpFP, RngIntElt -> GrpFPElt`
The $i$-th defining generator for $G$. A negative subscript indicates that the inverse of the generator is to be created. The generator `G.0` is `Identity(G)`, the empty word in $G$.
##### `Generators(G): GrpFP -> { GrpFPElt }`
A set containing the generators for the group $G$.
##### `NumberOfGenerators(G): GrpFP -> RngIntElt`
##### `Ngens(G): GrpFP -> RngIntElt`
The number of generators for the group $G$.
##### `PresentationLength(G): GrpFP -> RngIntElt`
The total length of the relators for $G$.
##### `Relations(G): GrpFP -> [ GrpFPRel ]`
A sequence containing the defining relations for $G$.
#### Operations on Words
The functions described in this section perform low level string operations like substitution, elimination, or substring matching on elements of fp-groups.
##### `Eliminate(u, x, v): GrpFPElt, GrpFPElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$, and a generator $x$, all belonging to a group $G$, return the word obtained from $u$ by replacing each occurrence of $x$ by $v$ and each occurrence of $x^{-1}$ by $v^{-1}$.
##### `Eliminate(U, x, v): { GrpFPElt }, GrpFPElt, GrpFPElt -> { GrpFPElt }`
Given a set of words $U$, a word $v$, and a generator $x$, all belonging to a group $G$, return the set of words obtained by taking each element $u$ of $U$ in turn, and replacing each occurrence of $x$ in $u$ by $v$ and each occurrence of $x^{-1}$ by $v^{-1}$.
##### `Match(u, v, f): GrpFPElt, GrpFPElt, RngIntElt -> BoolElt, RngIntElt`
Suppose $u$ and $v$ are words belonging to the same group $G$, and that $f$ is an integer such that $1 \leq f \leq \# u$. The function seeks the least integer $l$ such that:
**(a)**
$l\geq f$; and
**(b)**
$v$ appears as a subword of $u$, starting at the $l$-th letter of $u$.
If such an integer $l$ is found `Match` returns the value `true` and $l$. If no such $l$ is found, `Match` returns the value `false`.
##### `RotateWord(u, n): GrpFPElt, RngIntElt -> GrpFPElt`
The word obtained by cyclically permuting the word $u$ by $n$ places. If $n$ is positive, the rotation is from left to right, while if $n$ is negative the rotation is from right to left. In the case where $n$ is zero, the function returns $u$.
##### `Substitute(u, f, n, v): GrpFPElt, RngIntElt, RngIntElt, GrpFPElt -> GrpFPElt`
Given words $u$ and $v$ belonging to a group $G$, and non-negative integers $f$ and $n$, this function replaces the substring of $u$ of length $n$, starting at position $f$, by the word $v$. Thus, if $u = x_{i_1}^{e_1}\cdots x_{i_f}^{e_f}\cdots x_{i_{f+n-1}}^{e_{f+n-1}}\cdots x_{i_m}^{e_m}$ then the substring $x_{i_f}^{e_f}\cdots x_{i_{f+n-1}}^{e_{f+n-1}}$ is replaced by $v$. If the function is invoked with $v =$ Id(G), then the substring $x_{i_f}^{e_f}\cdots x_{i_{f+n-1}}^{e_{f+n-1}}$ of $u$ is deleted.
##### `Subword(u, f, n): GrpFPElt, RngIntElt, RngIntElt -> GrpFPElt`
The subword of the word $u$ comprising the $n$ consecutive letters commencing at the $f$-th letter of $u$.
##### `Example: Operations on Words in a Free Group (ex-1c9bc2)`
We demonstrate some of these operations in the context of the free group on generators $x$, $y$, and $z$.
```magma
> F := FreeGroup(3);
> u := (x, y*z);
> w := u^(x^2*y);
> #w;
12
> w;
y^-1 * x^-3 * z^-1 * y^-1 * x * y * z * x^2 * y
```
We replace each occurrence of the generator $x$ in $w$ by the word $y*z^-1$.
```magma
> Eliminate(w, x, y*z^-1);
y^-1 * z * y^-1 * z * y^-1 * z * y^-1 * z^-2 * y * z * y * z^-1 * y * z^-1 * y
```
We count the number of occurrences of each generator in $w$.
```magma
> [ ExponentSum(w, F.i) : i in [1..Ngens(F)] ];
[ 0, 0, 0 ]
> GeneratorNumber(w);
-2
```
We locate the start of the word $u$ in the word $w$.
```magma
> b, p := Match(w, u, 1);
> b, p;
true 4
```
We now replace the subword $u$ in $w$ by the word $y*x$.
```magma
> t := Substitute(w, p, #u, y*x);
> t;
y^-1 * x^-2 * y * x^3 * y
```
We create the set of all distinct cyclic permutations of the word $u$.
```magma
> rots := { RotateWord(u, i) : i in [1 ..#u] };
> rots;
{ y^-1 * x * y * z * x^-1 * z^-1, x * y * z * x^-1 * z^-1 * y^-1,
x^-1 * z^-1 * y^-1 * x * y * z, z * x^-1 * z^-1 * y^-1 * x * y,
z^-1 * y^-1 * x * y * z * x^-1, y * z * x^-1 * z^-1 * y^-1 * x }
```
#### Operations on Presentations
The functions described in this section construct a new fp-group from an existing one by adding or deleting a generator or by adding, deleting, or changing a relation. The new group is created without any relationship to the existing group.
##### `AddGenerator(G): GrpFP -> GrpFP`
Given an fp-group $G$ with presentation $\langle X\ |\ R \rangle$, create a new fp-group with presentation $\langle X \cup \{ z\rbrace\ |\ R \rangle$, where $z$ is a symbol not in $X$.
##### `AddGenerator(G, w): GrpFP, GrpFPElt -> GrpFP`
Given an fp-group $G$ with presentation $\langle X\ |\ R \rangle$, and given also a word $w$ in the generators $X$, create a new fp-group having the presentation $\langle X\ \cup \{ z\rbrace\ |\ R \cup \{ z = w \rbrace \rangle$, where $z$ is a symbol not in $X$.
##### `AddRelation(G, r): GrpFP, RelElt -> GrpFP`
Given an fp-group $G$, and a relation $r$ on the generators of $G$, create a new fp-group whose presentation consists of the relations of $G$ together with the relation $r$.
##### `AddRelation(G, g): GrpFP, GrpFPElt -> GrpFP`
Given an fp-group $G$, and an element $g$ of $G$, create a new fp-group whose presentation consists of the relations of $G$ together with the relation $g={\rm Id}(G)$.
##### `AddRelation(G, r, i): GrpFP, RelElt, RngIntElt -> GrpFP`
Given an fp-group $G$, and a relation $r$ on the generators of $G$, create a new fp-group which has as its presentation the relations of $G$ together with the relation $r$ inserted after the $i$-th existing relation of $G$.
##### `AddRelation(G, g, i): GrpFP, GrpFPElt, RngIntElt -> GrpFP`
Given an fp-group $G$, and an element $g$ of $G$, create a new fp-group which has as its presentation the relations of $G$ together with the relation $g={\rm Id}(G)$ inserted after the $i$-th existing relation of $G$.
##### `DeleteGenerator(G, x): GrpFP, GrpFPElt -> GrpFP`
Given an fp-group $G$ with presentation $\langle X\ |\ R \rangle$, and given also an element $z$ in $X$, create a new fp-group with presentation $\langle X \setminus \{z\}\ |\ R' \rangle$, where the relations $R'$ are obtained from $R$ by deleting all relations containing an occurrence of $z$.
##### `DeleteRelation(G, r): GrpFP, RelElt -> GrpFP`
Given an fp-group $G$, which includes the relation $r$ amongst its relations, create a new fp-group which has as its presentation the relations of $G$ with relation $r$ omitted.
##### `DeleteRelation(G, g): GrpFP, GrpFPElt -> GrpFP`
Given an fp-group $G$, which includes the relation $g={\rm Id}(G)$ amongst its relations, create a new fp-group which has as its presentation the relations of $G$ with this relation omitted.
##### `DeleteRelation(G, i): GrpFP, RngIntElt -> GrpFP`
Given an fp-group $G$, create a new fp-group which has as its presentation the relations for $G$ with the $i$-th relation deleted.
##### `ReplaceRelation(G, s, r): GrpFP, RelElt, RelElt -> GrpFP`
##### `ReplaceRelation(G, h, r): GrpFP, GrpFPElt, RelElt -> GrpFP`
##### `ReplaceRelation(G, s, g): GrpFP, RelElt, GrpFPElt -> GrpFP`
##### `ReplaceRelation(G, h, g): GrpFP, GrpFPElt, GrpFPElt -> GrpFP`
Given an fp-group $G$, which includes the relation $s$ or $h={\rm Id}(G)$ amongst its relations, create a new fp-group which has as its presentation the relations for $G$ with the relation $s$ replaced by the relation $r$ or $g={\rm Id}(G)$.
##### `ReplaceRelation(G, i, r): GrpFP, RngIntElt, RelElt -> GrpFP`
Given an fp-group $G$ and a relation $r$ in the generators of $G$, create a new fp-group which has as its presentation the relations for $G$ with relation number $i$ replaced by the relation $r$.
##### `ReplaceRelation(G, i, g): GrpFP, RngIntElt, GrpFPElt -> GrpFP`
Given an fp-group $G$ and an element $g$ of $G$, create a new fp-group which has as its presentation the relations for $G$ with relation number $i$ replaced by the relation $g={\rm Id}(G)$.
##### `Example: Replacing a Relation (ex-c80efd)`
We use the function [`ReplaceRelation`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#function-grpfp-replacerelation) to vary a particular relation in a presentation. The order of the resulting group together with the index of a particular subgroup is determined.
```magma
> G := FPGroup< x, y, z, h, k, a |
> x^2, y^2, z^2, (x,y), (y,z), (x,z), h^3, k^3, (h,k),
> (x,k), (y,k), (z,k), x^h*y, y^h*z, z^h*x, a^2, a*x*a*y,
> a*y*a*x, (a,z), (a,k), (a*h)^2 >;
> for i := 0 to 1 do
> for j := 0 to 1 do
> for k := 0 to 1 do
> for l := 0 to 2 do
> rel := G.1^i*G.2^j*G.3^k*G.5^l*(G.6*G.4)^2 = Id(G);
> K := ReplaceRelation(G, 21, rel);
> print Order(K), Index(K, sub< K | K.6, K.4>);
> end for;
> end for;
> end for;
> end for;
<0, 0, 0, 0> 144 24
<0, 0, 0, 1> 144 8
<0, 0, 0, 2> 144 8
<0, 0, 1, 0> 18 3
<0, 0, 1, 1> 18 1
<0, 0, 1, 2> 18 1
<0, 1, 0, 0> 72 3
<0, 1, 0, 1> 72 1
<0, 1, 0, 2> 72 1
<0, 1, 1, 0> 36 6
<0, 1, 1, 1> 36 2
<0, 1, 1, 2> 36 2
<1, 0, 0, 0> 18 3
<1, 0, 0, 1> 18 1
<1, 0, 0, 2> 18 1
<1, 0, 1, 0> 144 6
<1, 0, 1, 1> 144 2
<1, 0, 1, 2> 144 2
<1, 1, 0, 0> 36 6
<1, 1, 0, 1> 36 2
<1, 1, 0, 2> 36 2
<1, 1, 1, 0> 72 12
<1, 1, 1, 1> 72 4
<1, 1, 1, 2> 72 4
```
### Simplification
Simplification may be achieved by setting some parameters and trusting that the built-in strategy will produce the result desired. Alternatively the user may create a Tietze process allowing the user to interactively perform the simplification.
#### Automatic Simplification
Subgroups of finitely presented groups constructed in certain ways may be have a generating set containing redundant generators. The most important case in which this situation may occur is a subgroup of the domain of a homomorphism $f$ of groups, defined as the preimage under $f$ of some given subgroup of the codomain of $f$. In this case, the generating set of the preimage will contain generators of the kernel of $f$ and is likely to be not minimal.
Since reducing the generating set may be expensive and is not necessary in all situations, a reduction is not done automatically. Instead, Magma provides a function for reducing generating sets of finitely presented groups.
##### `ReduceGenerators(G): GrpFP -> GrpFP, Map`
Given a finitely presented group $G$, attempt to construct a presentation $H$ on fewer generators. $H$ is returned as a subgroup of $G$ (which of course is equal to $G$), so that element coerce is possible. The isomorphism from $G$ to $H$ is returned as second return value.
If a presentation for $G$ is known, this function attempts to simplify this presentation (cf. section [Interactive Simplification](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#grpfp-1-subsectiontietze)). Otherwise, it tries to rewrite $G$ with respect to a suitable supergroup to obtain a presentation on fewer generators.
For a sample application of this function, see Example [KG-Modules of a Finitely Presented Group](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/kg-modules.md#example-ex-00eade).
Given a finitely presented group $G$, the user can attempt to simplify its presentation using Tietze transformations and substring searching. The choice of simplification strategy can either be left to Magma or selected by the user.
##### `Simplify(G: parameters): GrpFP -> GrpFP, Map`
```magma
Preserve : [RngIntElt] Default: []
Iterations : RngIntElt Default: 10000
EliminationLimit : RngIntElt Default: 100
LengthLimit : RngIntElt Default: Infinity()
ExpandLimit : RngIntElt Default: 150
GeneratorsLimit : RngIntElt Default: 0
SaveLimit : RngIntElt Default: 10
SearchSimultaneous: RngIntElt Default: 20
Print : RngIntElt Default: 0
```
Given a finitely presented group $G$, attempt to simplify the presentation of $G$ by repeatedly eliminating generators and subsequently shortening relators by substitution of substrings that correspond to the left or right hand side of a relation. The order in which transformations are applied is determined by a set of heuristics. The transformation process terminates when no more eliminations of generators and no more length reducing substring replacements are possible. A new group $K$ isomorphic to $G$ is returned as a subgroup of $G$ which is (hopefully) defined by a simpler presentation than $G$. The isomorphism $f:G\rightarrow K$ is returned as second return value.
The simplification process can be controlled by a set of parameters described below.
The strategy employed by the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) can be controlled using the following set of parameters.
This parameter can be used to indicate that certain generators of $G$ should not be eliminated (*default*: no restrictions). `Preserve` is assigned a sequence of integers in the range $[1,\dots,n]$, where $n$ is the number of generators of $G$, containing the numbers of those generators of $G$ which should be preserved. See Example [Reducing a Generating Set](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#example-ex-df6881) for a sample application.
This parameter sets the maximal number of iterations of the main elimination / simplification cycle which will be performed.
This parameter sets the maximal number of generators which may be eliminated in each elimination phase.
If `LengthLimit` is set to $n$, any eliminations which would make the total length of the relators grow beyond $n$ will not be performed (*default*: no limit).
If `ExpandLimit` is set to $n$, the total length of the relators is not permitted to grow by more than a factor of $n\%$ in any elimination phase (*default*: 150%). If this limit is reached, the elimination phase is aborted.
Any eliminations which would reduce the number of generators below the value of `GeneratorsLimit` will not be performed (*default*: 0).
If `SaveLimit` is set to $n$, any simplification phase is repeated, if the reduction in the total length of the relators achieved during this phase exceeds $n\%$ (*default*: 10%).
This parameter sets the number of relators processed simultaneously in a simplification phase.
This parameter controls the volume of printing. By default its value is that returned by [`GetVerbose`](https://docs.magma-maths.org/MagmaLanguage/EnvironmentAndOptions/verbose.md#function-env-getverbose)`("Tietze")`, which is $0$ unless it has been changed through use of [`SetVerbose`](https://docs.magma-maths.org/MagmaLanguage/EnvironmentAndOptions/verbose.md#procedure-env-setverbose).
##### `Example: Simplifying the Fibonacci Group F(8) (ex-6d0638)`
Consider the Fibonacci group $F(8)$.
```magma
> F := FreeGroup(8);
> F8 :=
> quo< F | x1*x2=x3, x2*x3=x4, x3*x4=x5, x4*x5=x6,
> x5*x6=x7, x6*x7=x8, x7*x8=x1, x8*x1=x2>;
```
We use the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) in order to obtain a presentation of $F(8)$ on two generators.
```magma
> H<[y]>, f := Simplify(F8);
> H;
Finitely presented group H on 2 generators
Generators as words in group F8
y[1] = x3
y[2] = x4
Relations
y[2] * y[1]^-2 * y[2] * y[1]^-1 * y[2]^2 * y[1] * y[2]^2 *
y[1]^-1 = Id(H)
y[1] * y[2] * y[1] * y[2]^2 * y[1] * y[2] * y[1]^2 * y[2]^-1
* y[1] = Id(H)
```
The isomorphism $f$ can be used to express the old generators in terms of the new ones.
```magma
> f;
Mapping from: GrpFP: F8 to GrpFP: H
> f(x1);
y[1]^2 * y[2]^-1
```
##### `SimplifyLength(G: parameters): GrpFP -> GrpFP, Map`
Given a finitely presented group $G$, attempt to eliminate generators and shorten relators by locating substrings that correspond to the left or right hand side of a relation. The order in which transformations are applied is determined by a set of heuristics. As opposed to the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify), this function terminates the transformation process when the total length of the presentation starts to increase with the elimination of further generators.
A new group $K$ isomorphic to $G$ is returned which is (hopefully) defined by a simpler presentation than $G$. $K$ is returned as a subgroup of $G$. The isomorphism $f:G\rightarrow K$ is returned as second return value. This function accepts the same set of parameters as the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify).
#### Interactive Simplification
The user may exercise fine control over the simplification of a presentation by creating a Tietze process which allows the user to look at the result of a simplification operation before specifying the next step.
##### `TietzeProcess(G: parameters): GrpFP -> Process(Tietze)`
```magma
Preserve : [RngIntElt] Default: []
Iterations : RngIntElt Default: 10000
EliminationLimit : RngIntElt Default: 100
LengthLimit : RngIntElt Default: Infinity()
ExpandLimit : RngIntElt Default: 150
GeneratorsLimit : RngIntElt Default: 0
SaveLimit : RngIntElt Default: 10
SearchSimultaneous: RngIntElt Default: 20
Print : RngIntElt Default: 0
```
Create a Tietze process that takes the presentation for the fp-group $G$ as its starting point. This process may now be manipulated by various procedures that will be described below.
These parameters define the defaults used for the Tietze operations. Each of the various procedures described below allows some or all of these control parameters to be overridden.
For the meanings of the parameters, see the description under [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) above.
##### `ShowOptions(~P : parameters): GrpFPTietzeProc`
Display the defaults associated with the Tietze process $P$. The current status of all the control parameters may be viewed by using this function.
##### `SetOptions(~P : parameters): GrpFPTietzeProc`
Change the defaults associated with the Tietze process $P$. All of the control parameters may be overridden permanently by using this function.
##### `Simplify(~P : parameters): GrpFPTietzeProc`
##### `SimplifyPresentation(~P : parameters): GrpFPTietzeProc`
Use the default strategy to simplify the presentation as much as possible. The transformation process is terminated when no more eliminations of generators and no more length reducing substring replacements are possible. All the control parameters may be overridden for this function.
##### `SimplifyLength(~P : parameters): GrpFPTietzeProc`
Use the default strategy to simplify the presentation as much as possible. The transformation process is terminated when the total length of the presentation starts to increase with the elimination of further generators. All the control parameters may be overridden for this function.
##### `Eliminate(~P: parameters): GrpFPTietzeProc`
##### `EliminateGenerators(~P: parameters): GrpFPTietzeProc`
```magma
Relator : RngIntElt Default: 0
Generator: RngIntElt Default: 0
```
Eliminate generators in the presentation defined by the Tietze process $P$ under the control of the parameters. First any relators of length one are used to eliminate trivial generators. Then, if there are any non-involutory relators of length two, the generator with the higher number is eliminated. Of the control parameters, only `EliminationLimit, ExpandLimit, GeneratorsLimit`, and `LengthLimit` may be overridden by this function.
If $n > 0$, try to eliminate a generator using the $n$-th relator. If no generator is specified by the parameter `Generator` below, then the generator which is eliminated will be the one occurring once in the relator that produced the smallest total relator length.
If $n > 0$, try to eliminate the $n$-th generator. If no relation is specified by the parameter `Relator` above, then the shortest relator in which the $n$-th generator occurs exactly once (if any) is used.
##### `Search(~P: parameters): GrpFPTietzeProc`
Simplifies the presentation by repeatedly searching for common substrings in pairs of relators where the length of the substring is greater than half the length of the shorter relator and making the corresponding transformations. Relators of length 1 or 2 are also used to generate simplifications. The control parameters `SaveLimit` and `SearchSimultaneous` can be overridden.
##### `SearchEqual(~P: parameters): GrpFPTietzeProc`
Modifies the presentation by repeatedly searching for common substrings in pairs of relators where the length of the substring is exactly half the length of the shorter relator and making the corresponding transformations. The control parameter `SearchSimultaneous` can be overridden.
##### `Group(P): GrpFPTietzeProc -> GrpFP, Map`
Extract the group $G$ defined by the current presentation associated with the Tietze process $P$, together with the isomorphism between the original group and $G$. $G$ is returned as a subgroup of the original group underlying $P$.
##### `NumberOfGenerators(P): GrpFPTietzeProc -> RngIntElt`
##### `Ngens(P): GrpFPTietzeProc -> RngIntElt`
The number of generators for the presentation currently stored by the Tietze process $P$.
##### `NumberOfRelations(P): GrpFPTietzeProc -> RngIntElt`
##### `Nrels(P): GrpFPTietzeProc -> RngIntElt`
The number of relations in the presentation currently stored by the Tietze process $P$.
##### `PresentationLength(P): GrpFPTietzeProc -> RngIntElt`
The sum of the lengths of the relators in the presentation currently stored by the Tietze process $P$.
##### `Example: Interactive Simplification of the Fibonacci Group F(7) (ex-1eeac7)`
The Fibonacci group $F(n)$ is generated by $\{ x_1, \ldots, x_n \rbrace$ with defining relations
$$
x_ix_{i+1} = x_{i+2}, i \in \{ 1, \ldots, n \rbrace,
$$
where the subscripts are taken modulo $n$. Consider the Fibonacci group $F(7)$, which is defined in terms of the presentation
$$
\begin{aligned}.\end{aligned}
$$
The following code will produce a 2-generator, 2-relator presentation for $F(7)$:
```magma
> F := FreeGroup(7);
> F7 :=
> quo< F | x1*x2=x3, x2*x3=x4, x3*x4=x5, x4*x5=x6,
> x5*x6=x7, x6*x7=x1, x7*x1=x2 >;
> P := TietzeProcess(F7);
> for i := 7 to 3 by -1 do
> Eliminate(~P: Generator := i);
> end for;
> Search(~P);
> H, f := Group(P);
> H;
Finitely presented group H on 2 generators
Generators as words in group F7
x = x1
y = x2
Relations
x^-1 * y^-1 * x^-1 * y^-2 * x^-1 * y * x^-1 * y * x^-1 = Id(H)
x * y^3 * x^-1 * y * x^-1 * y^2 * x * y * x * y^-1 = Id(H)
```
The resulting presentation is
$$
.
$$
We can use the isomorphism $f$ returned by the function [`Group`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-tietze-group) to express arbitrary words in the original generators of $F(7)$ in terms of the new generators $x$ and $y$:
```magma
> f;
Mapping from: GrpFP: F7 to GrpFP: H
> f(x7);
x * y^2 * x * y^2 * x * y * x * y^2 * x * y
```
Alternatively, a similar effect may be obtained using the [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) function:
```magma
> F := FreeGroup(7);
> F7 :=
> quo< F | x1*x2=x3, x2*x3=x4, x3*x4=x5, x4*x5=x6,
> x5*x6=x7, x6*x7=x1, x7*x1=x2>;
> H, f := Simplify(F7: Iterations := 5);
> H;
Finitely presented group H on 2 generators
Generators as words in group F7
x = x2
y = x3
Relations
x * y^-1 * x * y^2 * x * y * x^2 * y^-1 = Id(H)
y * x * y^2 * x^-1 * y * x^-2 * y * x^-2 = Id(H)
```
Again, we can use the isomorphism $f$ returned by the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) to express arbitrary words in the original generators of $F(7)$ in terms of the new generators $x$ and $y$:
```magma
> f;
Mapping from: GrpFP: F7 to GrpFP: H
> f(x7);
y * x * y * x * y^2 * x * y
```
##### `Example: Reducing a Generating Set (ex-df6881)`
In a situation where some proper subset $S$ of the original generating set of a finitely group $G$ is sufficient to generate $G$, the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) can also be used to rewrite words in the original generators in terms of the elements of $S$. Consider again one of the Fibonacci groups, say $F(8)$.
```magma
> F := FreeGroup(8);
> F8 :=
> quo< F | x1*x2=x3, x2*x3=x4, x3*x4=x5, x4*x5=x6,
> x5*x6=x7, x6*x7=x8, x7*x8=x1, x8*x1=x2>;
```
Obviously, $F(8)$ is generated by $x_1$ and $x_2$. We utilise the function [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) to obtain a presentation $H$ of $F(8)$ on $x_1$ and $x_2$, using the parameter `Preserve` to indicate that $x_1$ and $x_2$ – i.e. the first and the second generator – are to be retained in the new presentation. We also compute the isomorphism $f:F(8)\rightarrow H$.
```magma
> H, f := Simplify(F8: Preserve := [1,2]);
```
Mapping elements of $F(8)$ to $H$ using the map $f$ basically means to rewrite these elements in terms of the generators $x = x_1$ and $y = x_2$. Since $H$ is returned as a subgroup of $F(8)$, the resulting words can be coerced from $H$ back into $F(8)$, yielding words explicitly in $x_1$ and $x_2$.
```magma
> F8 ! f(x5*x6);
x1 * x2^2 * x1 * x2 * x1^2 * x2^-1 * x1 * x2^-1
```
### Standard Constructions
In this section various intrinsics for constructing fp-groups are described. These are familiar groups constructed as fp-groups, extensions of existing fp-groups, and the construction of fp-groups from some other type of group.
#### Familiar Groups as FP-Groups
A number of functions are provided which construct presentations for various standard groups.
##### `AbelianFPGroup([n₁,...,nᵣ]): [ RngIntElt ] -> GrpFP`
Construct the abelian group defined by the sequence $[n_1,\ldots,n_r]$ of non-negative integers as an fp-group. The function returns the direct product of cyclic groups $C_{n_1} \times C_{n_2} \times \cdots \times C_{n_r}$, where $C_0$ is interpreted as an infinite cyclic group.
##### `AlternatingFPGroup(n): RngIntElt -> GrpFP`
##### `Alt(GrpFP, n): Cat, RngIntElt -> GrpFP`
Construct the alternating group of degree $n$ as an fp-group, where the generators correspond to the permutations $(3,4,\ldots ,n)$ and $(1,2,3)$, for $n$ odd, or $(1,2)(3,4$, $\ldots$, $n)$ and $(1,2,3)$, for $n$ even.
##### `BraidFPGroup(n): RngIntElt -> GrpFP`
Construct the braid group on $n$ strings ($n-1$ Artin generators) as an fp-group.
##### `CoxeterFPGroup(t): MonStgElt -> GrpFP`
Construct the Coxeter group of Cartan type $t$ as a finitely presented group, given by the standard Coxeter presentation. The Cartan type $t$ is passed to this function as a string; we refer to Chapter [Root Data](https://docs.magma-maths.org/LieTheory/RootData/index-root-data.md#chaprootdtm) for details.
##### `CyclicFPGroup(n): RngIntElt -> GrpFP`
Construct the cyclic group of order $n$ as an fp-group.
##### `DihedralFPGroup(n): RngIntElt -> GrpFP`
For $n > 2$, return the dihedral group of order $2n$ as an fp-group. For $n=0$, return the infinite dihedral group as an fp-group.
##### `ExtraSpecialFPGroup(p, n : parameters): RngIntElt, RngIntElt -> GrpFP`
```magma
Type: MonStgElt Default: "+"
```
Given a small prime $p$ and a small positive integer $n$, construct an extra-special group $G$ of order $p^{2n+1}$ in the category `GrpFP`. The isomorphism type of $G$ can be selected using the parameter `Type`.
Possible values for this parameter are “$+$” (default) and “$-$“.
If `Type` is set to “$+$”, the function returns for $p = 2$ the central product of $n$ copies of the dihedral group of order 8, and for $p > 2$ it returns the unique extra-special group of order $p^{2n+1}$ and exponent $p$.
If `Type` is set to “$-$”, the function returns for $p = 2$ the central product of a quaternion group of order 8 and $n-1$ copies of the dihedral group of order 8, and for $p > 2$ it returns the unique extra-special group of order $p^{2n+1}$ and exponent $p^2$.
##### `SymmetricFPGroup(n): RngIntElt -> GrpFP`
##### `Sym(GrpFP, n): Cat, RngIntElt -> GrpFP`
Construct the symmetric group of degree $n$ as an fp-group, where the generators correspond to the permutations $(1,2,\ldots,n)$ and $(1,2)$.
##### `Example: Standard Groups as FP-Groups (ex-936a87)`
We create the symmetric group ${\operatorname{Sym}}(8)$ as an fp-group:
```magma
> S8 := SymmetricFPGroup(8);
> S8;
Finitely presented group S8 on 2 generators
Relations
S8.1^8 = Id(S8)
S8.2^2 = Id(S8)
(S8.1 * S8.2)^7 = Id(S8)
(S8.1^-1 * S8.2 * S8.1 * S8.2)^3 = Id(S8)
(S8.2 * S8.1^-2 * S8.2 * S8.1^2)^2 = Id(S8)
(S8.2 * S8.1^-3 * S8.2 * S8.1^3)^2 = Id(S8)
(S8.2 * S8.1^-4 * S8.2 * S8.1^4)^2 = Id(S8)
```
We create the Coxeter group of Cartan type $F4$ as an fp-group:
```magma
> F := CoxeterFPGroup("F4");
> F;
Finitely presented group F on 4 generators
Relations
(F.2 * F.3)^2 = (F.3 * F.2)^2
F.1^2 = Id(F)
F.1 * F.3 = F.3 * F.1
F.2 * F.4 = F.4 * F.2
F.1 * F.2 * F.1 = F.2 * F.1 * F.2
F.2^2 = Id(F)
F.3^2 = Id(F)
F.3 * F.4 * F.3 = F.4 * F.3 * F.4
F.4^2 = Id(F)
F.1 * F.4 = F.4 * F.1
```
#### Construction of Extensions
##### `Darstellungsgruppe(G): GrpFP -> GrpFP`
Given an fp-group $G$, construct a maximal central extension $\tilde G$ of $G$. The group $\tilde G$ is created as an fp-group.
##### `DirectProduct(G, H): GrpFP, GrpFP -> GrpFP`
Given two fp-groups $G$ and $H$, construct the direct product of $G$ and $H$.
##### `DirectProduct(Q): [ GrpFP ] -> GrpFP`
Given a sequence $Q$ of $r$ fp-groups, construct the direct product $Q[1] \times \dots \times Q[r]$.
##### `FreeProduct(G, H): GrpFP, GrpFP -> GrpFP`
Given two fp-groups $G$ and $H$, construct the free product of $G$ and $H$.
##### `FreeProduct(Q): [ GrpFP ] -> GrpFP`
Given a sequence $Q$ of $r$ fp-groups, construct the free product of the groups $Q[1], \dots, Q[r]$.
##### `Example: Maximal Central Extension of a Group (ex-dc16bb)`
We construct a maximal central extension of the following group of order $36$.
```magma
> G := FPGroup;
> G;
Finitely presented group G on 2 generators
Relations
x1^4 = Id(G)
(x1 * x2^-1)^2 = Id(G)
x2^4 = Id(G)
(x1 * x2)^3 = Id(G)
> D := Darstellungsgruppe(G);
> D;
Finitely presented group D on 4 generators
Relations
D.1^4 * D.3^-1 * D.4^2 = Id(D)
D.1 * D.2^-1 * D.1 * D.2^-1 * D.4 = Id(D)
D.2^4 = Id(D)
D.1 * D.2 * D.1 * D.2 * D.1 * D.2 * D.4 = Id(D)
(D.1, D.3) = Id(D)
(D.2, D.3) = Id(D)
(D.1, D.4) = Id(D)
(D.2, D.4) = Id(D)
(D.3, D.4) = Id(D)
> Index(D, sub< D | >);
108
```
Thus, a maximal central extension of $G$ has order $108$
##### `Example: Direct Product of FP-Groups (ex-bdaee4)`
We create the direct product of the alternating group of degree $5$ and the cyclic group of order $2$.
```magma
> A5 := FPGroup;
> Z2 := quo< FreeGroup(1) | $.1^2 >;
> G := DirectProduct(A5, Z2);
> G;
Finitely presented group G on 3 generators
Relations
G.1^2 = Id(G)
G.2^3 = Id(G)
(G.1 * G.2)^5 = Id(G)
G.3^2 = Id(G)
G.1 * G.3 = G.3 * G.1
G.2 * G.3 = G.3 * G.2
```
#### Conversion to FP-Groups
In this section we describe intrinsics which given a group in some category other than that of fp-group, return an isomorphic fp-group.
##### `FPGroup(G): GrpPerm -> GrpFP, Hom(Grp)`
##### `FPGroup(G): GrpMat -> GrpFP, Hom(Grp)`
Given a finite group $G$ in category `GrpPerm` or `GrpMat`, this function returns a finitely presented group $F$, isomorphic to $G$, together with the isomorphism $\phi: F\rightarrow G$. The generators of $F$ correspond to the generators of $G$, so this function can be used to obtain a set of defining relations for the given generating set of $G$.
It should be noted that this function is only practical for groups of order at most a few million. In the case of much larger permutation groups, an isomorphic fp-group can be constructed using the function [`FPGroupStrong`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-fpgroupstrong).
##### `Example: FP-Group from a Permutation Group (ex-185d54)`
We define the alternating group $G \simeq A_{10}$ as a permutation group.
```magma
> G := Alt(10);
> G;
Permutation group G acting on a set of cardinality 5
Order = 1814400 = 2^7 * 3^4 * 5^2 * 7
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10)
(1, 2, 3)
```
Now we create an fp-group $F$ isomorphic to $G$, using the function [`FPGroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-fpgroup). The presentation is constructed by computing a set of defining relations for the generators of $G$, i.e. the generators of the returned fp-group correspond to the generators of $G$. This defines a homomorphism from $F$ to $G$, which the function [`FPGroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-fpgroup) returns as second return value.
```magma
> F, f := FPGroup(G);
> F;
Finitely presented group F on 2 generators
Relations
x^8 = Id(F)
y^3 = Id(F)
(x^-1, y^-1)^2 = Id(F)
(x * y^-1 * x^-2 * y^-1 * x)^2 = Id(F)
(x * y * x^-3 * y^-1 * x^2)^2 = Id(F)
x^-1 * y^-1 * x^-4 * y^-1 * x^4 * y^-1 * x^4 * y^-1 * x^-3 = Id(F)
(y^-1 * x^-1)^9 = Id(F)
> f;
Mapping from: GrpFP: F to GrpPerm: G
> f(x);
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10)
> f(y);
(1, 2, 3)
```
Constructing a nice presentation for a large finite permutation group or matrix group is a difficult problem. The next two intrinsics obtain a presentation as a side effect of the Todd-Coxeter-Schreier-Sims algorithm.
##### `FPGroupStrong(G): GrpPerm -> GrpFP, Hom(Grp)`
##### `FPGroupStrong(G): GrpMat -> GrpFP, Hom(Grp)`
Given a finite group $G$ in category [`GrpPerm`](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm) or [`GrpMat`](https://docs.magma-maths.org/FiniteGroups/MatrixGroupsOverGeneralRings/index-matrix-groups-over-general-rings.md#chapgrpmat), this function returns a finitely presented group $F$, isomorphic to $G$, together with the isomorphism $\phi:F \ar G$. The generators of $F$ correspond to a set of strong generators of $G$. If no strong generating set is known for $G$, one will be constructed.
For a detailed description of this function, in particular for a list of available parameters, we refer to Chapter [Permutation Groups](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm) and Chapter [Matrix Groups over General Rings](https://docs.magma-maths.org/FiniteGroups/MatrixGroupsOverGeneralRings/index-matrix-groups-over-general-rings.md#chapgrpmat), respectively.
##### `FPGroupStrong(G, N): GrpPerm, GrpPerm -> GrpFP, Hom(Grp)`
Given a permutation group $G$ and a normal subgroup $N$ of $G$, this function returns a finitely presented group $F$, isomorphic to $G/N$, together with a homomorphism $\phi:G \rightarrow F$.
For a detailed description of this function, we refer to Chapter [Permutation Groups](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm).
##### `Example: FP-Group on Strong Generators (ex-f1a62b)`
We take the alternating group $G \simeq A_5$ as a permutation group.
```magma
> G := Alt(5);
> G;
Permutation group G acting on a set of cardinality 5
Order = 60 = 2^2 * 3 * 5
(1, 2)(3, 4, 5, 6, 7, 8, 9, 10)
(1, 2, 3)
```
Using the function [`FPGroupStrong`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/standard.md#function-grpfp-1-fpgroupstrong), we now create an fp-group $Fs$, isomorphic to $G$, whose generators correspond to a set of strong generators of $G$.
```magma
> Fs<[z]>, fs := FPGroupStrong(G);
> Fs;
Finitely presented group Fs on 3 generators
Relations
z[1]^-3 = Id(Fs)
(z[1]^-1 * z[3]^-1)^2 = Id(Fs)
z[3]^-3 = Id(Fs)
z[1]^-1 * z[2]^-1 * z[1] * z[3]^-1 * z[2] = Id(Fs)
z[2]^-3 = Id(Fs)
(z[3]^-1 * z[2]^-1)^2 = Id(Fs)
> fs;
Mapping from: GrpFP: Fs to GrpPerm: G
```
Applying the isomorphism $fs$, we have a look at the strong generating set constructed for $G$.
```magma
> [ fs(z[i]) : i in [1..#z] ];
[
(3, 4, 5),
(1, 2, 3),
(2, 3, 4)
]
```
It should be noted that `FPGroupStrong` constructs a presentation on what is often a much larger generating set and consequently the presentation can be much larger. However, this intrinsic will produce presentations for very large groups as compared to the intrinsic `FPGroup`. It is recommended that the user apply the simplification algorithm (intrinsic `Simplify`) to the presentation returned by `FPGroupStrong`).
Groups that satisfy certain properties, such as being abelian or polycyclic, are known to possess presentations with respect to which the word problem is soluble. Specialised categories have been constructed in Magma for several of these, e.g. the categories [`GrpGPC`](https://docs.magma-maths.org/FinitelyPresentedGroups/PolycyclicGroups/index-polycyclic-groups.md#chapgrpgpc), [`GrpPC`](https://docs.magma-maths.org/FiniteGroups/FiniteSolubleGroups/index-finite-soluble-groups.md#chapgrppc) and [`GrpAb`](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab). The functions described in this section allow a group created in one of the special presentation categories to be recast as an fp-group.
##### `FPGroup(G): GrpPC -> GrpFP, Hom(Grp)`
##### `FPGroup(G): GrpGPC -> GrpFP, Hom(Grp)`
##### `FPGroup(G): GrpAb -> GrpFP, Hom(Grp)`
Given a group $G$, defined either by a polycyclic group presentation (types `GrpPC` and `GrpGPC`) or an abelian group presentation (type `GrpAb`), return a group $H$ isomorphic to $G$, together with the isomorphism $\phi:G \rightarrow H$. The generators for $H$ will correspond to the generators of $G$. The effect of this function is to convert a presentation in a special form into a general fp-group presentation.
##### `Example: Converting a Polycyclic Group to an FP-Group (ex-a91a69)`
We illustrate the cast from special forms of fp-groups to the category `GrpFP` by converting a polycyclic group.
```magma
> G := DihedralGroup(GrpGPC, 0);
> G;
GrpGPC : G of infinite order on 2 PC-generators
PC-Relations:
G.1^2 = Id(G),
G.2^G.1 = G.-2
> F := FPGroup(G);
> F;
Finitely presented group F on 2 generators
Relations
F.1^2 = Id(F)
F.2^F.1 = F.2^-1
F.1^-1 * F.2^-1 * F.1 = F.2
```
There is a special Magma category [`GrpPermCox`](https://docs.magma-maths.org/LieTheory/CoxeterGroups/index-coxeter-groups.md#chapgrppermcox), a subcategory of [`GrpPerm`](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm), for finite Coxeter groups. Here, we describe a function to create from a Coxeter group $W$ a finitely presented group $F$, isomorphic to $W$, which is given by the standard Coxeter group presentation. We refer to Chapter [Coxeter Groups](https://docs.magma-maths.org/LieTheory/CoxeterGroups/index-coxeter-groups.md#chapgrppermcox) for the details.
##### `CoxeterFPGroup(W): GrpFPCox -> GrpFP, Map`
##### `CoxeterFPGroup(W): GrpPermCox -> GrpFP, Map`
```magma
Local: BoolElt Default: false
```
Given a finite Coxeter group $W$ in the category [`GrpFPCox`](https://docs.magma-maths.org/LieTheory/CoxeterGroups/index-coxeter-groups.md#chapgrpfpcox) or [`GrpPermCox`](https://docs.magma-maths.org/LieTheory/CoxeterGroups/index-coxeter-groups.md#chapgrppermcox), construct a finitely presented group $F$ isomorphic to $W$, given by a standard Coxeter presentation. The isomorphism from $W$ to $F$ is returned as second return value. The first argument to this function must be the category `GrpFP`.
If the parameter `Local` is set to `true`, $F$ is the appropriate subgroup of the FP version of the overgroup of $W$.
##### `Example: Coxeter Group as an FP-Group (ex-1b08d9)`
We construct a Coxeter group $W$ of Cartan type $C5$ and create an isomorphic fp-group $F$. We can use the isomorphism from $W$ to $F$ to map words in the generators of $F$ to permutation group elements and vice versa.
```magma
> W := CoxeterGroup("C5");
> F<[s]>, h := CoxeterFPGroup(W);
> F;
Finitely presented group F on 5 generators
Relations
s[3]^2 = Id(F)
s[2] * s[4] = s[4] * s[2]
s[1]^2 = Id(F)
s[1] * s[2] * s[1] = s[2] * s[1] * s[2]
s[2] * s[5] = s[5] * s[2]
s[4]^2 = Id(F)
s[1] * s[3] = s[3] * s[1]
s[3] * s[4] * s[3] = s[4] * s[3] * s[4]
s[2]^2 = Id(F)
s[1] * s[4] = s[4] * s[1]
s[5]^2 = Id(F)
(s[4] * s[5])^2 = (s[5] * s[4])^2
s[3] * s[5] = s[5] * s[3]
s[1] * s[5] = s[5] * s[1]
s[2] * s[3] * s[2] = s[3] * s[2] * s[3]
> h;
Mapping from: GrpCox: W to GrpFP: F given by a rule
> h(W.1*W.2);
s[1] * s[2]
> (s[1]*s[2]*s[3]*s[4]) @@ h;
(1, 39, 4, 3, 2)(5, 13, 19, 23, 25)(6, 36, 35, 8, 7)(9, 16, 21,
24, 17)(10, 33, 32, 31, 11)(12, 18, 22, 15, 20)(14, 29, 28,
27, 26)(30, 38, 44, 48, 50)(34, 41, 46, 49, 42)(37, 43, 47,
40, 45)
```
### Properties of an FP-Group
In this section intrinsics for providing information about the global properties of an fp-group are described.
#### Cardinality
A very basic requirement is to determine whether a given fp-group $G$ is finite or infinite. Unfortunately, this is a problem for which there is no general solution. Instead, an intrinsic `Order` is provided which attempts to prove that the group is finite and a second intrinsic `IsInfiniteFPGroup` is provided which attempts to prove that the group is infinite. If both fail to produce a result then no conclusion can be drawn.
We assume that the group is finite and try to prove that this assumption is true using coset enumeration and other techniques.
##### `Order(G: parameters): GrpFP -> RngIntElt`
##### `FactoredOrder(G: parameters): GrpFP -> [ ]`
Given an fp-group $G$, this function attempts to determine the order of $G$ or to prove that $G$ is infinite. If a finite order can be computed, the function `Order` returns the order as a positive integer, whereas the function `FactoredOrder` returns a sequence of prime power factors. The function `FactoredOrder` reports an error in all other cases, whereas the function `Order` returns the object `Infinity`, if $G$ can be shown to be infinite and returns a value of 0 if neither a finite value for the group order nor a proof for the infinity of $G$ can be obtained. No conclusions can be drawn from a return value 0 of `Order`.
In addition to the parameters controlling possibly invoked coset enumerations, there exist some other parameters controlling the strategy used by the functions `Order` and `FactoredOrder`. These parameters are described below.
##### `Example: Order of a Finitely Presented Group (ex-134948)`
We use the function [`Order`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-properties.md#function-grpfp-1-order) without any parameters to compute the order of the group
$$
G =\,\, .
$$
```magma
> G := FPGroup;
> G;
Finitely presented group G on 2 generators
Relations
x^8 = Id(G)
y^7 = Id(G)
(x * y)^2 = Id(G)
(x^-1 * y)^3 = Id(G)
> Order(G);
10752
```
The strategy employed by the functions `Order` and `FactoredOrder` may involve trying to obtain information on certain subgroups of $G$. Whether or not an attempt is made to construct a presentation for a subgroup arising in the course of the computation by means of Reidemeister-Schreier rewriting, is controlled by three parameters:
```magma
UseRewrite: BoolElt Default: true
MinIndex : RngIntElt Default: 10
MaxIndex : RngIntElt Default: 1000
```
If `UseRewrite` is set to `false`, attempts to construct presentations for subgroups are not made. Otherwise, `MinIndex` and `MaxIndex` specify for subgroups of which index range Reidemeister-Schreier rewriting is done.
The following strategy is used for trying to determine the order of $G$.
**(1)**
Check whether $G$ is free. If so, $G$ is either trivial or infinite.
**(2)**
Check whether the presentation for $G$ is deficient (i.e. whether the number of relations is smaller than the number of generators). If it is, $G$ is infinite.
**(3)**
Check the subgroups of $G$ with known order. If such a subgroup is known to be infinite or if we can compute its index in G, we’re done.
**(4)**
Try to compute the index of $G$ in a supergroup of known order. (An infinite supergroup in which $G$ has finite index proves $G$ to be infinite.)
**(5)**
Try to enumerate the cosets of the trivial subgroup in $G$.
**(6)**
Check the subgroups of known or easily computable index in $G$. If we can compute the order of such a subgroup or prove that it is infinite, we’re done.
**(7)**
Try to enumerate the cosets of some subgroups occurring “naturally” in the presentation of $G$.
**(8)**
Check the supergroups in which $G$ has known or easily computable index. If we can compute the order of a supergroup or prove that it is infinite, we’re done.
**(9)**
Try to rewrite $G$ w.r.t. some supergroup and to enumerate the cosets of the trivial subgroup using the resulting presentation.
Steps requiring coset enumeration in $G$ or a supergroup of $G$ are skipped, if no relations are known for this group. Steps involving Reidemeister-Schreier rewriting may be skipped according to the values of the parameters mentioned above.
Experienced users can control the behaviour of coset enumerations which may be invoked by the functions `Order` and `FactoredOrder` with a wide range of parameters. Both functions – in addition to the parameters mentioned above – accept the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
The intrinsic `IsInfiniteFPGroup` attempts to prove that an fp-group is infinite by finding a subgroup that has an infinite abelian section.
##### `IsInfiniteFPGroup(G : parameters): GrpFP -> BoolElt`
Given a finitely-presented group $G$, the function attempts to establish that $G$ is infinite. If it succeeds it returns `true`. Otherwise it returns `false` meaning that the tests it has made don’t prove the group to be infinite. This does not imply that the group is likely to be finite. The code searches for epimorphisms $\phi$ of $G$ and uses the Holt-Plesken cohomological criterion, as implemented in the intrinsic `HasPositiveH1Dimension`, to test for non-finiteness of the kernel of $\phi$. The version in this release works best on groups $G$ which are perfect or which have small abelian quotients. It will be extended to handle groups which have larger soluble quotients in a subsequent release.
At present epimorphisms are sought using four techniques that are described elsewhere in this chapter: L2-quotient, Simple Quotient, Low Index Subgroups, and Low Index Normal Subgroups. The intrinsic has corresponding parameters `L2Quot`, `SimQuot`, `LISub`, and `LINSub` that allow the user to control the types of epimorphisms to be used. The default is for all four methods to be employed – however, the user can turn any of them off by setting the appropriate parameter to `false`. A verbose flag `InfGrp` will provide some information about progress.
The application of the Holt-Plesken theorem to an epimorphism $\phi$ requires a knowledge of the $QH$ modules, where $H$ is the image of $\phi$. A database containing such modules for ATLAS groups of degree up to 1000 is available as an optional download. The serious user of `IsInfiniteFPGroup` is strongly advised to have this database available as it can save a great deal of runtime.
##### `Example: Proving a Group Infinite (ex-fea44f)`
Consider the following one-relator quotient group of the triangle group $\langle 2, 3, 7 \rangle$:
$$
\langle x,y | x^2, y^3, (xy)^7, (x,y)^{12} \rangle
$$
```magma
> SetVerbose("InfGrp", 2);
> G := FPGroup;
> inf := IsInfiniteFPGroup(G);
>
L2Q SEARCH commencing:
L2Q SEARCH Considering L2Qs with degrees in the interval [1, 1000]
SIMPLE QUOTIENT Commence searching interval [60, 10000]:
Found epi onto PSL(2, 7) of order 168 and degree 8: 0.000 secs.
Found epi onto PSL(2, 13) of order 1092 and degree 14: 0.060 secs.
SIMPLE QUOTIENT Search of interval [60, 10000] unsuccessful: 2 epis, 0.610
secs.
NORMAL SUBGROUPS Search of indices [1, 1000] commencing:
NORMAL SUBGROUPS Search of [1, 1000] unsuccessful: 1 + 0 subgroups, 0.200
secs.
SUBGRP SEARCH of indices [1, 25] commencing:
SUBGRP SEARCH of [1, 25] was unsuccessful: 8 + 8 subgroups, 0.990 secs.
SIMPLE QUOTIENT Commence searching interval [20001, 100000]:
SIMPLE QUOTIENT Search of interval [20001, 100000] unsuccessful: 0 epis,
0.620 secs.
NORMAL SUBGROUPS Search of indices [1001, 10000] commencing:
NORMAL SUBGROUPS Search of [1001, 10000] unsuccessful: 0 + 0 subgroups,
0.960 secs.
SUBGRP SEARCH of indices [26, 50] commencing:
SUBGRP SEARCH of [26, 50] was unsuccessful: 8 + 8 subgroups, 1.460 secs.
SIMPLE QUOTIENT Commence searching interval [100001, 1000000]:
Found epi onto J_2 of order 604800 and degree 100: 1.300 secs.
For 4950-dim ex_sq of perm module, dim of H^1/Z is 1: 1.050 secs.
SIMPLE QUOTIENT Search of interval [100001, 1000000] successful: 1 epis,
2.430 secs.
> inf;
true
```
So the group $G$ has the simple group $J_2$ as quotient and the rational representation corresponding to the exterior square of $J_2$’s degree 100 permutation module lifted to $G$ has first cohomology group of dimension 1. So $G$ is infinite by an application of the Holt-Plesken theorem.
##### `HasPositiveH1Dimension(G, phi : parameters): GrpFP, HomGrp -> BoolElt`
Given a finitely-presented group $G$ and an epimorphism $\phi$ of $G$ onto a permutation group $H$, this intrinsic returns `true` if some ${{\mathbb{Q}}}H$-module, where ${{\mathbb{Q}}}$ is the field of the rationals, lifts to a ${{\mathbb{Q}}}G$-module whose first cohomology group is non-trivial. By a result of Holt and Plesken this shows that $G$ is infinite. The ${{\mathbb{Q}}}H$-modules are obtained in three ways. The first consists of ${{\mathbb{Q}}}H$-modules that are easily obtainable from the permutation representation of $H$. The second method is to systematically construct the ${{\mathbb{Q}}}H$-modules of $H$ while the third source is the Magma database of rational representations of small simple groups. Parameters allow the user to specify which of these methods are used (see below).
The parameter `ESLimit` allows the user to place a limit on the dimension of exterior squares for which the Holt-Plesken test will be applied. If the parameter `CmpIrrs` is set to `true` then Magma will construct rational irreducible representations for $H$ if they are not present in the database. If the parameter `DBIrrs` is set to `true` then Magma will use those irreducible modules for $H$ that are available in the Rational Representations Database. Setting both `CmpIrrs` and `DBIrrs` to `false` specifies that the test should be applied only to the permutation module and its exterior square.
There are two signatures for this intrinsic where the second has a third argument needed by the intrinsic `IsInfiniteFPGroup`. Note that this version is only intended to be called internally. For all other uses the two-argument version should be used.
#### Small Cancellation Conditions
##### `SmallCancellationConditions(G): GrpFP -> RngIntElt, RnIntElt, FldRatElt)`
```magma
Print : RngIntElt Default: 0
involgens: BoolElt Default: false
```
This functions calculates the values for which the small cancellation conditions $T(i)$, $C(j)$, and $C'(k)$ are satisfied by the presentation $G$. It returns $i,j,k'$, where $T(i)$, $C(j)$, and $C'(k)$ are satisfied for all $k > k'$.
If the parameter `involgens` is set `true` then edges labelled by involutory generators $x$ in a van Kampen diagram for $G$ are regarded as undirected, and are not regarded as pieces of the relator $x^2$.
##### `Example: Small Cancellation Conditions (ex-2f799a)`
```magma
> G := FPGroup< x,y | x^6, y^6, (x*y)^5, (x,y)^3 >;
> SmallCancellationConditions(G);
This presentation satisfies T( 3 ).
This presentation satisfies C( 5 ).
This presentation satisfies C'(k) for all k > 1/5 .
3 5 1/5
```
#### Largeness
##### `IsLarge(G, L, U:parameters): GrpFP, RngIntElt, RngIntElt -> BoolElt, GrpFP`
```magma
Simplify : BoolElt Default: true
MaxExponent: RngIntElt Default: 100000
TimeLimit : RngIntElt Default: 10
```
Attempt to show that $G$ has a subgroup with homomorphic image the free group of rank 2. If successful return `true`, plus a subgroup witness to this property. If not successful, return `false`. Note that `false` does not imply that $G$ doesn’t have the stated property, just that no such subgroup has been found. The witness group returned has a finite index subgroup with the free group of rank 2 as a homomorphic image. The algorithm will consider subgroups of $G$ with index $h$ having $L\le h\le U$.
We attempt to decide if G is large by calculating the (multivariable) Alexander polynomial of those subgroups $H$ of $G$ having index $h$ in the given range, and also subgroups of H having index up to $2h$ in $G$.
Controls simplification of subgroup presentations.
If the abelian torsion of subgroup $H$ is greater than `MaxExponent` we do not rewrite to obtain a presentation of $H$.
If, in computing low index subgroups of a finite index subgroup $H$ of $G$, the low-index subgroup calculation takes more than `TimeLimit` in seconds, then abort without trying to find more subgroups of $H$.
### Subgroups
#### Specification of a Subgroup
##### `sub< G | L >: GrpFP, List -> GrpFP`
Construct the subgroup $H$ of the fp-group $G$ generated by the words specified by the terms of the *generator list* $L = L_1, \ldots, L_r$.
A term $L_i$ of the generator list may consist of any of the following objects:
**(a)**
A word;
**(b)**
A set or sequence of words;
**(c)**
A sequence of integers representing a word;
**(d)**
A set or sequence of sequences of integers representing words;
**(e)**
A subgroup of an fp-group;
**(f)**
A set or sequence of subgroups.
The collection of words and groups specified by the list must all belong to the group $G$ and $H$ will be constructed as a subgroup of $G$.
The generators of $H$ consist of the words specified directly by terms $L_i$ together with the stored generating words for any groups specified by terms of $L_i$. Repetitions of an element and occurrences of the identity element are removed (unless $H$ is trivial).
If the `sub`-constructor is invoked with an empty list $L$, the trivial subgroup will be constructed.
##### `sub< G | f >: GrpFP, Hom(Grp) -> GrpFP`
Given a homomorphism $f$ from $G$ onto a transitive subgroup of Sym($n$), construct the subgroup of $G$ which affords this permutation representation.
##### `ncl< G | L >: GrpFP, List -> GrpFP`
Construct the subgroup $N$ of the fp-group $G$ as the normal closure of the subgroup $H$ generated by the words specified by the terms of the *generator list* $L$.
The possible forms of a term of the generator list are the same as for the [`sub< G | L >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructor-grpfp-1-sub)-constructor.
This constructor may be applied even when $H$ has infinite index in $G$, provided that its normal closure $N$ has finite index. The subgroup $N$ is obtained by computing the coset table of the trivial subgroup in the group defined by the relations of $G$ together with relators corresponding to the words generating $H$. For a sample application of this function, see Example [Braid Group onto Sym(5)](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#example-ex-63663e).
This function may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, see Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `ncl: GrpFP, Hom(Grp) -> GrpFP`
Given a homomorphism $f$ from $G$ onto a transitive subgroup of Sym($n$), construct the subgroup of $G$ that is the normal closure of the subgroup $K$ of $G$ which affords this permutation representation.
##### `CommutatorSubgroup(G): GrpFP -> GrpFP`
##### `DerivedSubgroup(G): GrpFP -> GrpFP`
##### `DerivedGroup(G): GrpFP -> GrpFP`
Given an fp-group $G$, try to construct the derived subgroup $G^\prime$ of $G$ as finite index subgroup of $G$. The construction fails if no presentation for $G$ is known or can be constructed, or if the index of $G^\prime$ in $G$ is too large or infinite.
This function may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, see Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Example: Subgroup of Index 448 (ex-6af151)`
The group $(8, 7\ |\ 2, 3)$ is defined by the presentation
$$
< a, b \ | \ a^8, b^7, (ab)^2, (a^{-1}b)^3 >,
$$
and has a subgroup of index 448 generated by the words $a^2$ and $a^{-1}b$:
```magma
> G := FPGroup;
> G;
Finitely presented group G on 2 generators
Relations
a^8 = Id(G)
b^7 = Id(G)
(a * b)^2 = Id(G)
> H := sub< G | a^2, a^-1 * b >;
> H;
Finitely presented group H on 2 generators
Generators as words in group G
x = a^2
y = a^-1 * b
```
##### `Example: Subgroup Defined by a Homomorphism (ex-4fd961)`
Given the group $G$ defined by the presentation
$$
< a, b \ | \ a^8, b^7, (ab)^2, (a, b)^9 >,
$$
there is a homomorphism into ${\operatorname{Sym}}(9)$ defined by
$$
\begin{aligned}a &\rightarrow (2, 4)(3, 5)(6, 7)(8, 9)\\
b& \rightarrow (1, 2, 3)(4, 6, 7)(5, 8, 9)\end{aligned}
$$
We construct the subgroup $H$ of $G$ that is the preimage of the stabiliser of the point $1$ in $G$.
```magma
> G := FPGroup< a, b | a^2, b^3, (a*b)^7, (a, b)^9>;
> T := PermutationGroup< 9 | (2, 4)(3, 5)(6, 7)(8, 9),
> (1, 2, 3)(4, 6, 7)(5, 8, 9) >;
> f := hom< G -> T | a -> T.1, b ->T.2 >;
> H := sub< G | f >;
> H;
Finitely presented group H
Subgroup of group G defined by coset table
> Index(G, H);
9
```
Using the function [`GeneratingWords`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-generatingwords), we obtain a set of generators for $H$.
```magma
> print GeneratingWords(G, H);
{ a, b^-1 * a * b^3 * a * b, b * a * b * a * b * a * b^-1,
b^3, b^-1 * a * b * a * b * a * b, b * a * b^3 * a * b^-1 }
```
#### The Todd-Coxeter Algorithm
This section describes the simplest use of coset enumeration techniques in Magma. Magma also provides interactive facilities for coset enumeration. For information on these more advanced uses, the user is referred to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
The Todd-Coxeter implementation installed in Magma is based on the stand-alone coset enumeration programme `ACE3` developed by George Havas and Colin Ramsay at the University of Queensland. The reader should consult [[CDHW73](https://docs.magma-maths.org/references.md#cite-cdhw-todd-coxeter)] and [[Hav91](https://docs.magma-maths.org/references.md#cite-havas-todd-coxeter)] for an explanation of the terminology and a general description of the algorithm. A manual for `ACE3` as well as the sources of ACE3 can be found online [[Ram](https://docs.magma-maths.org/references.md#cite-ace-homepage)].
Experienced users can control the Todd-Coxeter procedures invoked by the functions described in this section with a wide range of parameters. For a complete description of these parameters and their meanings we refer to the manual entry for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess). We just mention briefly the most important ones:
```magma
CosetLimit: RngIntElt Default: 0
```
If `CosetLimit` is set to $n$, where $n$ is a positive integer, then the coset table may have at most $n$ rows. In other words, a maximum of $n$ cosets can be defined at any instant during the enumeration. It is ensured in this case, that enough memory is allocated to store the requested number of cosets, regardless of the value of the parameter [`Workspace`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#grpfp-1-workspace). If `CosetLimit` is set to 0 (default), the maximal number of active cosets is determined by the size of the coset table (cf. parameter [`Workspace`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#grpfp-1-workspace)) and the number of columns of the coset table (i.e. the number of group generators).
```magma
Workspace: RngIntElt Default: 4000000
```
The number of words allocated for the coset table. Note that if [`CosetLimit`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#grpfp-1-cosetlimit) is set, at least as much memory is allocated as is necessary to store the requested number of cosets.
```magma
Strategy: MonStgElt Default:
```
Using this parameter one of several predefined strategies can be selected. (See the manual entry for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration) for a complete list.) The most important values of this parameter are:
**$\;\;\bullet$**
“Easy”: This selects a combination of parameters, which in situations where an overflow is not expected is likely to produce a result more quickly and using less memory than the default strategy. This strategy will fail for more complicated enumerations.
**$\;\;\bullet$**
“Hard”: This selects a combination of parameters, which is more likely to produce a finite result for difficult coset enumerations than the default strategy. This strategy will use more memory and enumerations usually will take longer.
##### `ToddCoxeter(G, H: parameters): GrpFP, GrpFP -> RngIntElt, Map, RngIntElt, RngIntElt`
Given a subgroup $H$ of the fp-group $G$, this function attempts to build up a coset table of $H$ in $G$ using the Todd-Coxeter procedure.
The first return value is the index of $H$ in $G$ (or 0, if the enumeration fails to complete). The other values returned are the coset table map, the maximum number of simultaneously active cosets, and the total number of cosets defined during the enumeration.
Experienced users can control the Todd-Coxeter procedure invoked by this function with a wide range of parameters. This function accepts the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Index(G, H: parameters): GrpFP, GrpFP -> RngIntElt`
##### `FactoredIndex(G, H: parameters): GrpFP, GrpFP -> [ ]`
Given a subgroup $H$ of the fp-group $G$, these functions attempt to determine the index of $H$ in $G$ by enumerating the cosets of $H$ using the Todd-Coxeter procedure. The index is returned as a positive integer (`Index`) or as a sequence of factors (`FactoredIndex`), respectively. If the coset enumeration fails to complete with a closed coset table, `Index` returns a value of 0, whereas `FactoredIndex` reports an error. No conclusion can be drawn in this case.
Experienced users can control the Todd-Coxeter procedure invoked by these functions with a wide range of parameters. Both functions accept the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Example: Todd-Coxeter Enumeration of 448 Cosets (ex-0e7ed5)`
The classical test example for Todd-Coxeter programmes is the enumeration of the 448 cosets of the subgroup $H =\,\, $ in the group $G =\,\, $.
```magma
> F := FreeGroup(2);
> G := quo;
> H := sub;
> Index(G, H);
448
```
##### `Example: Todd-Coxeter for the Harada-Norton Group (ex-c75263)`
The Harada-Norton simple group has the presentation
$$
\begin{aligned}< x, a, b, c, d, e, f, g\ | \ & x^2, a^2, b^2, c^2, d^2, e^2, f^2, g^2, (x,a), (x,g),\\
& (bc)^3, (bd)^2, (be)^2, (bf)^2, (bg)^2, (cd)^3, (ce)^2, (cf)^2, (cg)^2,\\
& (de)^3, (df)^2, (dg)^2, (ef)^3, (eg)^2, (fg)^3, (b, xbx), (a, edcb),\\
& (a,f)dcbdcd, (ag)^5, (cdef, xbx), (b,xcdefx), (cdef, xcdefx) >\end{aligned}
$$
The subgroup generated by $x, b, c, d, e, f, g$ has index 1,140,000. We use the parameter [`CosetLimit`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) to request a sufficiently large coset table. For the enumeration we choose the predefined strategy [`Hard`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) with the modification of a complete C-style lookahead ([`Lookahead`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) `:= 2`).
```magma
> HN :=
> FPGroup< x, a, b, c, d, e, f, g |
> x^2, a^2, b^2, c^2, d^2, e^2, f^2, g^2,
> (x, a), (x, g),
> (b*c)^3, (b*d)^2, (b*e)^2, (b*f)^2, (b*g)^2,
> (c*d)^3, (c*e)^2, (c*f)^2, (c*g)^2,
> (d*e)^3, (d*f)^2, (d*g)^2,
> (e*f)^3, (e*g)^2,
> (f*g)^3,
> (b, x*b*x),
> (a, e*d*c*b), (a, f)*d*c*b*d*c*d, (a*g)^5,
> (c*d*e*f, x*b*x), (b, x*c*d*e*f*x),
> (c*d*e*f, x*c*d*e*f*x)
> >;
> H := sub;
> idx := Index(HN, H: Print := true, CosetLimit := 1200000,
> Strategy := "Hard", Lookahead := 2);
INDEX = 1140000
(a=1140000 r=1471 h=1168483 n=1168483;
l=2945 c=201.17;
m=1142416 t=1470356)
> idx;
1140000
```
##### `Example: Orders of a Parametrised Family of Groups (ex-4e7bab)`
We use a function representing a parametrised presentation to determine the order of a collection of groups obtained by systematically varying one relation. We select the predefined coset enumeration strategy [`Easy`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) for the order computations.
```magma
> Grp := func< p, q, r, s |
>
> FPGroup<
> x, y, z, h, k, a |
> x^2, y^2, z^2, (x,y), (y,z), (x,z), h^3, k^3, (h,k),
> (x,k), (y,k), (z,k), x^h*y, y^h*z, z^h*x, a^2, a*x*a*y,
> a*y*a*x, (a,z), (a,k), x^p*y^q*z^r*k^s*(a*h)^2 >
> >;
> [ < , Order(Grp(i,j,k,l) : Strategy := "Easy") >
> : i, j, k in [0..1], l in [0..2] ];
[ <<0, 0, 0, 0>, 144>, <<0, 0, 1, 0>, 18>, <<0, 1, 0, 0>, 72>,
<<0, 1, 1, 0>, 36>, <<1, 0, 0, 0>, 18>, <<1, 0, 1, 0>, 144>,
<<1, 1, 0, 0>, 36>, <<1, 1, 1, 0>, 72>, <<0, 0, 0, 1>, 144>,
<<0, 0, 1, 1>, 18>, <<0, 1, 0, 1>, 72>, <<0, 1, 1, 1>, 36>,
<<1, 0, 0, 1>, 18>, <<1, 0, 1, 1>, 144>, <<1, 1, 0, 1>, 36>,
<<1, 1, 1, 1>, 72>, <<0, 0, 0, 2>, 144>, <<0, 0, 1, 2>, 18>,
<<0, 1, 0, 2>, 72>, <<0, 1, 1, 2>, 36>, <<1, 0, 0, 2>, 18>,
<<1, 0, 1, 2>, 144>, <<1,1, 0, 2>, 36>, <<1, 1, 1, 2>, 72> ]
```
#### Interactive Coset Enumeration
##### Introduction
This section presents the interactive coset enumeration facility of Magma. This concept makes it possible to restart an enumeration after changing enumeration parameters or adding relators or subgroup generators, while making use of information obtained up to that point as much as possible. It is thus particularly suitable for very hard enumerations, requiring a careful and interactive choice of enumeration parameters or for series of similar enumerations.
The Todd-Coxeter implementation installed in Magma is based on the stand alone coset enumeration programme `ACE3` developed by George Havas and Colin Ramsay at the University of Queensland. The reader should consult [[CDHW73](https://docs.magma-maths.org/references.md#cite-cdhw-todd-coxeter)] and [[Hav91](https://docs.magma-maths.org/references.md#cite-havas-todd-coxeter)] for an explanation of the terminology and a general description of the algorithm. A manual for `ACE3` as well as the sources of `ACE3` can be found online [[Ram](https://docs.magma-maths.org/references.md#cite-ace-homepage)].
In Magma an interactive coset enumeration is realised as an object of the category `GrpFPCosetEnumProc` which can be created and modified, allows starting and restarting of coset enumerations and provides access to internal data like the coset table and various status information.
##### Constructing and Modifying a Coset Enumeration Process
###### `CosetEnumerationProcess(G, H: parameters): GrpFP, GrpFP -> GrpFPCosetEnumProc`
```magma
Compact : RngIntElt Default: 10
CosetLimit : RngIntElt Default: 0
Workspace : RngIntElt Default: 4000000
FillFactor : RngIntElt Default: 0
CTFactor : RngIntElt Default: 1000
RTFactor : RngIntElt Default: 2000/l
Style : MonStgElt Default: "R_CR"
Lookahead : RngIntElt Default: 0
Mendelsohn : BoolElt Default: false
RelationsInSubgroup: RngIntElt Default: -1
RowFilling : BoolElt Default: true
PrefDefMode : RngIntElt Default: 3
PrefDefSize : RngIntElt Default: 8
DeductionMode : RngIntElt Default: 4
DeductionSize : RngIntElt Default: 1000
PathCompression : BoolElt Default: false
TimeLimit : RngIntElt Default: -1
LoopLimit : RngIntElt Default: 0
LowerBound : RngIntElt Default: 1
Print : BoolElt Default: false
Messages : RngIntElt Default: 0
Strategy : MonStgElt Default:
```
This function creates a coset enumeration process for enumerating the cosets of the subgroup $H$ of the finitely presented group $G$. Note that no actual coset enumeration is started for the created coset enumeration process. This can be done with the function [`StartEnumeration`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-startenumeration).
The user can control in detail the way a subsequent enumeration will be performed with the help of the following parameters. For a more thorough explanation of the parameters and of how they affect the coset enumeration, we refer to the `ACE3` manual.
This parameter controls the compaction of the coset table during an enumeration. A compaction will be done if a new coset definition is required, there is no space for a new coset available in the coset table and the percentage of dead cosets exceeds the value of the parameter `Compact`.
If `CosetLimit` is set to $n$, where $n$ is a positive integer, then the coset table may have at most $n$ rows. In other words, a maximum of $n$ cosets can be defined at any instant during the enumeration. It is ensured in this case, that enough memory is allocated to store the requested number of cosets, regardless of the value of the parameter [`Workspace`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess). If `CosetLimit` is set to 0 (default), the maximal number of active cosets is determined by the size of the coset table (cf. parameter [`Workspace`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess)) and the number of columns of the coset table (i.e. the number of group generators).
The number of words allocated for the coset table. Note that if [`CosetLimit`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) is set, at least as much memory is allocated as is necessary to store the requested number of cosets.
In certain situations it is necessary to ensure that a certain proportion, the *fill fraction* (see Havas (1991)), of the coset table is always kept filled, even if *preferred definitions* are made. The parameter `FillFactor` allows the user to specify the fill fraction which is the reciprocal of `FillFactor`. The default value of 0 selects a fill factor of $\lfloor (5*(c+2))/4\rfloor$, where $c$ is the number of columns in the coset table.
These parameters control the enumeration style and the balance between coset table style definitions (C-style, Felsch style) and relator table style definitions (R-style, HLT style).
The parameters `CTFactor` and `RTFactor` set the number of C-style coset definitions and R-style coset applications, respectively, which are performed in every step of the appropriate type. The default value for `RTFactor` is $2000/l$, where $l$ is the total length of the relators.
The possible arguments to the parameter `Style` are `"R"`, `"C"`, `"R_CR"`, `"Rc"`, `"Cr"`, `"Rt"`, and `"CR"`.
In style R all definitions are made via relator scans, i.e. this is HLT mode.
In style C all definitions are made by filling the next empty position in the coset table and testing the new coset in all essentially different positions in the relators, i.e. this is Felsch mode.
In style R$\underline{\;\;}$CR we run in style R until an overflow occurs, perform a lookahead on the entire table and then switch to style CR (see below).
The styles Rc and Cr are like the styles R and C, except that a single style C or style R pass is done after the initial R or style C pass, respectively.
In style Rt definitions are made as in style R, but the new cosets are tested in all essentially different positions in the relators as in style C.
In style CR alternate passes of style C and style R are performed, with all definitions tested in all essentially different positions in the relators as in style C.
In HLT style enumerations, possible implications of new definitions or deductions made while tracing relators may not be detected until much later. One possible solution to this problem is to perform lookaheads occasionally, i.e. to process the coset table, looking for deductions or coincidences. A lookahead can be done using the entire table (*complete*) or only that part of the table above the current coset (*partial*) and it can be done in R-style (scanning cosets from the beginning of relators) or in C-style (testing all definitions in all essentially different relator positions). The following lookahead modes are supported.
For `Lookahead := 0` (default) no lookahead is done.
For `Lookahead := 1` a partial R-style lookahead is done.
For `Lookahead := 2` a complete C-style lookahead is done.
For `Lookahead := 3` a complete R-style lookahead is done.
For `Lookahead := 4` a partial C-style lookahead is done.
If `Mendelsohn` is set, coset applications are done at all cyclic permutations of the relators. This is usually very expensive but may nevertheless be helpful in certain cases.
This parameter controls, whether relators are treated as additional subgroup generators, i.e. whether they are applied to coset 1 at the start of an enumeration. An argument of $-1$ (default) includes all relators, an argument of 0 turns this feature off and a positive argument includes the appropriate number of relators, in order.
In R-style it is usual to scan each row of the table after its coset has been applied to all relators, and to make definitions to fill any holes encountered. Failure to do so can cause even simple enumerations to overflow. To switch row filling off, set the parameter `RowFilling` to false.
If the argument is 0, then Felsch style definitions are made using the next empty position in the coset table. Otherwise, gaps of length one found during relator scans are preferentially filled. If the argument is 1, they are filled immediately, and if it is 2, the consequent deduction is also made immediately. If the argument is 3, then the gaps are noted in the preferred definition queue and the next coset definition will be made to fill the oldest gap of length one.
This parameter controls the size of the preferred definition queue, which is implemented as a ring buffer, dropping earliest entries. Setting `PrefDefSize` to $n$ allocates a buffer of size $2^n$.
A completed table is only valid if every table entry has been tested in all essentially different relator positions. Untested deductions are stored on a stack. This parameter allows the user to specify how deductions should be handled. The possible actions are:
For `DeductionMode := 0` : discard deductions if there is no stack space left.
For `DeductionMode := 1` : as 0, but redundant cosets are purged off the top of the stack whenever a coincidence is found.
For `DeductionMode := 2` : as 0, but all redundant cosets are purged from the stack whenever a coincidence is found.
For `DeductionMode := 3` : discard the entire stack if it overflows.
For `DeductionMode := 4` : if the stack overflows, then double the stack size and purge all redundant cosets from the stack.
If deductions are discarded for any reason during an enumeration, then a final relator application pass will be done at the end of the enumeration automatically to check the result.
Sets the (initial) size of the deduction stack in words, with one deduction taking two words. A value of 0 selects the default size of 1000 words.
Switching this option on reduces the amount of data movement during coincidence processing at the expense of tracing and compressing coincidence paths, which involves many coset table accesses. The value of this parameter has no effect on the result but may influence the running time.
This parameter puts a time limit in seconds on the length of an enumeration. A value of $-1$ (default) means no limit. A value of 0 performs exactly one pass through the main enumeration loop.
The core enumerator is organised as a state machine, with each step performing an action (e.g. lookahead, compaction) or a block of actions (e.g. [`CTFactor`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) coset definitions or [`RTFactor`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) coset applications). Using this parameter, a limit on the total number of steps can be imposed. A value of 0 (default) turns this feature off.
This may be set to any known lower bound for the index. Should it happen that the coset table has no holes, and the number of active cosets is equal to the given bound, the enumeration will terminate. When the given bound is equal to the index, this will save tracing relators at many cosets when there is no possibility of finding coincidences.
If this parameter is set to true, the enumerator prints a single message at the end of an enumeration and possibly some progress messages during an enumeration (cf. parameter [`Messages`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess)).
If the argument $n$ of this parameter is non-zero (and [`Print`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) is set to true), then a progress message is printed after every $|n|$ “actions” (i.e. coset definitions, deductions, coincidences, etc.) A negative value of $n$ turns hole monitoring on.
Using this parameter one of several predefined strategies can be selected. The effect is equivalent to selecting an appropriate combination of arguments for some of the parameters explained above. Any predefined strategy can be modified by explicitly overriding parameter values.
The following predefined strategies exist: “Default”, “Easy”, “Hard”, “Felsch”, “HLT”, “CT”, “RT”, “Sims1”, “Sims3”, “Sims5”, “Sims7”, “Sims9”.
All predefined strategies set [`PrefDefSize`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) `:= 8`, [`DeductionSize`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) `:= 1000`, [`PathCompression`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) `:= false` and [`LoopLimit`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) `:= 0`. For the remaining parameter settings see the tables below.
```
Default Easy Hard Felsch HLT CT
Compact 10 100 10 10 10 100
Workspace 4 1 10 4 4 4
FillFactor 0 1 0 0 1 1
CTFactor 1000 0 1000 1000 0 1000
RTFactor 2000/l 1000 1 0 1000 0
Style R_CR R CR C R C
Lookahead 0 0 0 0 1 0
Mendelsohn false false false false false false
RelationsInSubgroup -1 0 -1 -1 0 0
RowFilling true true true false true false
PrefDefMode 3 0 3 3 0 0
DeductionMode 4 0 4 4 0 4
RT Sims1 Sims3 Sims5 Sims7 Sims9
Compact 100 10 10 10 10 10
Workspace 4 4 4 4 4 4
FillFactor 1 1 1 1 1 1
CTFactor 0 0 0 0 0 1000
RTFactor 1000 1000 1000 1000 1000 0
Style R R Rt R Rt C
Lookahead 0 0 0 0 0 0
Mendelsohn false false false true true false
RelationsInSubgroup 0 0 0 0 0 0
RowFilling false true true true true false
PrefDefMode 0 0 0 0 0 0
DeductionMode 0 0 4 0 4 4
```
###### `AddRelator(~P, w): GrpFPCosetEnumProc, GrpFPElt`
Add a word $w$ in the generators of the group $G$ underlying the coset enumeration process $P$ to the defining relations of $G$. This means that a coset enumeration process $P$ for the cosets of $H$ in $G$ is transformed into a coset enumeration process for the cosets of $\pi(H)$ in $\pi(G)$, where $\pi: G \rightarrow \pi(G)$ is an epimorphism with kernel $\langle w\rangle^G$. (Here $\langle w\rangle^G$ denotes the normal closure in $G$ of the subgroup of $G$ generated by $w$.)
###### `AddSubgroupGenerator(~P, w): GrpFPCosetEnumProc, GrpFPElt`
Add an element $w$ of the group $G$ underlying the coset enumeration process $P$ to the generators of the subgroup. This means that a coset enumeration process $P$ for the cosets of $H$ in $G$ is transformed into a coset enumeration process for the cosets of $\langle H, w\rangle$ in $G$, where $\langle H, w\rangle$ denotes the subgroup of $G$ generated by $H$ and $w$.
###### `SetProcessParameters(~P: parameters): GrpFPCosetEnumProc`
Change enumeration parameters of the coset enumeration process $P$. The set of parameters accepted by this function is the same as for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess); see there for a description.
All parameters which are not explicitly changed or modified by selecting one of the predefined strategies retain their old values.
It should be noted that it is not possible to decrease the workspace allocated by a coset enumeration process, once an enumeration has been started. However the workspace can be extended without invalidating any information contained in the process.
##### Starting and Restarting an Enumeration
There are several ways of starting and restarting an enumeration for a coset enumeration process, which retain information from previous enumerations to a varying extent.
###### `StartEnumeration(~P: parameters): GrpFPCosetEnumProc`
Start a new enumeration for $P$. All information in $P$ is discarded. This function can be called at any time for an existing coset enumeration process. The enumeration parameters for $P$ can be modified by passing parameters to this function. (This is equivalent to calling the function [`SetProcessParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setprocessparameters) before calling `StartEnumeration`.) The set of parameters accepted by this function is the same as for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess); see there for a description.
###### `RedoEnumeration(~P: parameters): GrpFPCosetEnumProc`
Restart an enumeration for $P$. All information in $P$ is retained and the enumeration is restarted at coset number 1. This function can be called for any coset enumeration process, which contains a valid coset table. (If $P$ does not contain a valid coset table, a call to `RedoEnumeration` causes a runtime error. Use the function [`CanRedoEnumeration`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-canredoenumeration) to check whether a call to `RedoEnumeration` is legal for a certain coset enumeration process.) Note that the coset table of $P$ need not be complete to use this function.
This function is intended for the case where additional relators and/or subgroup generators have been introduced. The coset table contained in $P$ is still valid. However, the additional data may allow the enumeration to compete, or cause a collapse to a smaller index.
The enumeration parameters for $P$ can be modified by passing parameters to this function. (This is equivalent to calling the function [`SetProcessParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setprocessparameters) before calling `RedoEnumeration`.) The set of parameters accepted by this function is the same as for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess); see there for a description.
###### `CanRedoEnumeration(P): GrpFPCosetEnumProc -> BoolElt`
Returns `true`, if a call to [`RedoEnumeration`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-redoenumeration) is legal for the coset enumeration process $P$.
###### `ContinueEnumeration(~P: parameters): GrpFPCosetEnumProc`
Continue an enumeration for $P$, which has been interrupted because some limit was exceeded. All information in $P$ is retained and the enumeration is restarted at the coset where the previous enumeration was stopped. This function can only be called for a coset enumeration process, if the previous enumeration produced a valid coset table and the subgroup underlying $P$ has not been changed since then. (Otherwise, a call to `ContinueEnumeration` causes a runtime error. Use the function [`CanContinueEnumeration`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cancontinueenumeration) to check whether a call to `ContinueEnumeration` is legal for a certain coset enumeration process.) Note that the coset table of $P$ need not be complete to use this function.
This function is intended for the case where an enumeration stopped without producing a finite index. This function allows to continue the enumeration with modified enumeration parameters with the minimal possible overhead.
The enumeration parameters for $P$ can be modified by passing parameters to this function. (This is equivalent to calling the function [`SetProcessParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setprocessparameters) before calling `ContinueEnumeration`.) The set of parameters accepted by this function is the same as for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess); see there for a description.
###### `CanContinueEnumeration(P): GrpFPCosetEnumProc -> BoolElt`
Returns `true`, if a call to [`ContinueEnumeration`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-continueenumeration) is legal for the coset enumeration process $P$.
###### `ResumeEnumeration(~P: parameters): GrpFPCosetEnumProc`
Resume or start an enumeration for $P$ in the “cheapest” way permitted by the state of the coset enumeration process $P$. The call
```magma
ResumeEnumeration(~P : parameters);
```
is equivalent to
```magma
if CanContinueEnumeration(P) then
ContinueEnumeration(~P : parameters);
elif CanRedoEnumeration(P) then
RedoEnumeration(~P : parameters);
else
StartEnumeration(~P : parameters);
end if;
```
The enumeration parameters for $P$ can be modified by passing parameters to this function. (This is equivalent to calling the function [`SetProcessParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setprocessparameters) before calling `ResumeEnumeration`.) The set of parameters accepted by this function is the same as for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess); see there for a description.
##### Accessing Information
###### `CosetsSatisfying(P : parameters): GrpFPCosetEnumProc -> { GrpFPElt }`
###### `CosetSatisfying(P : parameters): GrpFPCosetEnumProc -> { GrpFPElt }`
```magma
First : RngIntElt Default: 1
Last : RngIntElt Default: 0
Limit : RngIntElt Default: 0
Normalizing: BoolElt Default: false
Order : RngIntElt Default: 0
Print : RngIntElt Default: 0
```
Given a coset enumeration process $P$ with underlying group $G$ and subgroup $H$, which contains a valid coset table, these functions return a set of words representing cosets which satisfy the conditions defined in the parameters.
A call to `CosetSatisfying` is equivalent to calling `CosetsSatisfying` with [`Limit`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying) set to 1 and [`First`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying) set to 2 (unless a higher value for [`First`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying) is specified explicitly), i.e. the function returns when the first coset, other than $H$, satisfying the specified conditions has been found.
This parameter determines at which coset the search for coset representatives satisfying the designated conditions is started. The coset $H$ always is coset number 1, i.e. setting `First` to 2 restricts the search to non-trivial cosets.
For the function `CosetSatisfying`, the minimal possible value for this parameter is 2; setting it to 1 does not cause an error but will be ignored.
This parameter determines at which coset the search for coset representatives satisfying the designated conditions is stopped. If it is set to 0 (default), all active cosets (starting from [`First`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying)) are searched.
If this parameter is set to $l > 0$, the search for coset representatives is aborted as soon as $l$ cosets satisfying the designated condition have been found. If it is set to 0 (default for `CosetsSatisfying`), no limit is in force.
This parameter is not available for the function `CosetSatisfying`.
If true, select coset representatives $x$ such that $x^{-1}h_ix$ is known to be contained in $H$ from the information in the coset table of $P$ for every generator $h_i$ of $H$. (i.e. $x$ is recognised as an element of the normaliser of $H$ in $G$.)
Select coset representatives $x$ such that $x^n$ is known to be contained in $H$ from the information in the coset table of $P$.
If the value of this parameter is positive, print the coset representatives found to satisfy the designated conditions.
These functions can be called for any coset enumeration process, which contains a valid coset table. If $P$ does not contain a valid coset table, a call to any of these functions causes a runtime error. You can use the function [`HasValidCosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable) to check whether a call to these functions is legal for a certain coset enumeration process.
###### `CosetTable(P): GrpFPCosetEnumProc -> Map`
The current coset table of $P$ as a map $f:\{1,\ldots,r\}\times G\rightarrow \{0,\ldots,r\}$, where $G$ and $H$ are the finitely presented group and the subgroup underlying $P$, respectively, and $r$ is number of active cosets. $f(i,x)$ is the coset to which coset $i$ is mapped under the action of $x\in G$. The value 0 is only included in the codomain if the coset table is not complete, and it denotes that the image of $i$ under $x$ is not known.
This function can be called for any coset enumeration process, which contains a valid coset table. If $P$ does not contain a valid coset table, a call to `CosetTable` causes a runtime error. Use the function [`HasValidCosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable) to check whether a call to `CosetTable` is legal for a certain coset enumeration process. Note that the coset table of $P$ need not be complete to use this function.
###### `HasValidCosetTable(P): GrpFPCosetEnumProc -> BoolElt`
Returns `true`, if $P$ contains a valid (but not necessarily closed) coset table, i.e. if a call to [`CosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosettable) is legal for the coset enumeration process $P$.
###### `HasClosedCosetTable(P): GrpFPCosetEnumProc -> BoolElt`
###### `HasCompleteCosetTable(P): GrpFPCosetEnumProc -> BoolElt`
Returns `true`, if $P$ contains a closed, valid coset table.
Note that if `HasClosedCosetTable` returns `true` for a coset enumeration process $P$, then in particular a call to [`CosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosettable) is legal for $P$.
###### `ExcludedConjugate(P): GrpFPCosetEnumProc -> GrpFPElt`
###### `ExcludedConjugates(P): GrpFPCosetEnumProc -> { GrpFPElt }`
Given a coset enumeration process $P$ with underlying group $G$ and subgroup $H$, which contains a valid coset table, these functions return a set $E$ containing either at most one word (`ExcludedConjugate`) or all words (`ExcludedConjugates`) of the form $g_i^{-1}h_jg_i$, where $g_i$ is a generator of $G$ and $h_j$ is a generator of $H$, such that $g_i^{-1}h_jg_i$ is not known to lie in $H$ from the information contained in the coset table of $P$.
If $E$ is empty, then $H$ is a normal subgroup of $G$. Otherwise the addition of elements of $E$ to the generators of $H$ may yield a larger subgroup of the normal closure of $H$ in $G$. In the case of a non-complete coset table it may happen, however, that excluded conjugates are found which actually lie in $H$.
These functions can be called for any coset enumeration process, which contains a valid coset table. If $P$ does not contain a valid coset table, a call to `ExcludedConjugate` or `ExcludedConjugates` causes a runtime error. You can use the function [`HasValidCosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable) to check whether a call to these functions is legal for a certain coset enumeration process.
###### `ExistsCosetSatisfying(P : parameters): GrpFPCosetEnumProc -> BoolElt, GrpFPElt`
Given a coset enumeration process $P$ with underlying group $G$ and subgroup $H$, which contains a valid coset table, return whether or not there exists a coset other than $H$, which satisfies the conditions defined in the parameters. If such a coset exists, a representing word is returned as second return value. This function accepts the same set of parameters as the function [`CosetSatisfying`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetsatisfying); see there for a description.
This function can be called for any coset enumeration process, which contains a valid coset table. If $P$ does not contain a valid coset table, a call to `ExistsCosetSatisfying` causes a runtime error. You can use the function [`HasValidCosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable) to check whether a call to `ExistsCosetSatisfying` is legal for a certain coset enumeration process.
###### `ExistsExcludedConjugate(P): GrpFPCosetEnumProc -> BoolElt, GrpFPElt`
Given a coset enumeration process $P$ with underlying group $G$ and subgroup $H$, which contains a valid coset table, return whether or not there exists a word of the form $g_i^{-1}h_jg_i$, where $g_i$ is a generator of $G$ and $h_j$ is a generator of $H$, such that $g_i^{-1}h_jg_i$ is not known to lie in $H$ from the information contained in the coset table of $P$. If the answer is positive, such a word is returned as second return value.
This function can be called for any coset enumeration process, which contains a valid coset table. If $P$ does not contain a valid coset table, a call to `ExistsExcludedConjugate` causes a runtime error. You can use the function [`HasValidCosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable) to check whether a call to `ExistsExcludedConjugate` is legal for a certain coset enumeration process.
Note that the coset table of $P$ need not be complete to call this function. A negative result of `ExistsExcludedConjugate` always implies that $H$ is a normal subgroup of $G$, even if the coset table of $P$ is not complete. In the case of a non-complete coset table it may happen, however, that excluded conjugates are found which actually lie in $H$.
###### `ExistsNormalisingCoset(P): GrpFPCosetEnumProc -> BoolElt, GrpFPElt`
###### `ExistsNormalizingCoset(P): GrpFPCosetEnumProc -> BoolElt, GrpFPElt`
Returns `true`, if an element of $G\setminus H$ which normalises $H$ can be found from the coset table contained in $P$. (Here, $G$ and $H$ are the finitely presented group and the subgroup underlying $P$, respectively.) If the answer is positive, such an element is returned as second return value. This function can be called for any coset enumeration process, which contains a valid coset table. If $P$ does not contain a valid coset table, a call to `ExistsNormalisingCoset` causes a runtime error. You can use the function [`HasValidCosetTable`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidcosettable) to check whether a call to `ExistsNormalisingCoset` is legal for a certain coset enumeration process.
Note that the coset table of $P$ need not be complete to call this function. However, no conclusion can be drawn from a negative result in the case of a non-complete coset table.
###### `Group(P): GrpFPCosetEnumProc -> GrpFP`
Returns the group underlying $P$ as a finitely presented group.
###### `Index(P): GrpFPCosetEnumProc -> RngIntElt`
Returns the index of $H$ in $G$. (Here, $G$ and $H$ denote the finitely presented group and the subgroup underlying $P$, respectively.)
This function can only be called, if the last enumeration done for $P$ has completed successfully with a finite index. Otherwise, a call to `Index` will cause a runtime error. Use the function [`HasValidIndex`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-hasvalidindex) to check whether a call to `Index` is legal for a certain coset enumeration process.
###### `HasValidIndex(P): GrpFPCosetEnumProc -> BoolElt`
Returns `true`, if the last enumeration done for $P$ has completed successfully with a finite index, i.e., if a call to [`Index`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-index) is legal for the coset enumeration process $P$.
###### `MaximalNumberOfCosets(P): GrpFPCosetEnumProc -> RngIntElt`
Returns the maximal number of cosets which were simultaneously active during the last enumeration done for $P$, or 1 if no enumeration has been done for $P$.
This function may be useful for assessing the performance of a certain set of enumeration parameters.
###### `Subgroup(P): GrpFPCosetEnumProc -> GrpFP`
Returns the subgroup $H$ underlying the coset enumeration process $P$. $H$ is returned as a subgroup of $G$, where $G$ is the finitely presented group underlying $P$.
###### `TotalNumberOfCosets(P): GrpFPCosetEnumProc -> RngIntElt`
Returns the total number of cosets defined during the last enumeration done for $P$, or 1 if no enumeration has been done for $P$.
This function may be useful for assessing the performance of a certain set of enumeration parameters.
###### `Example: Interactive Coset Table for Harada-Norton (ex-bfc054)`
In the Harada-Norton sporadic simple group
$$
\begin{aligned}< x, a, b, c, d, e, f, g\ | \
& x^2, a^2, b^2, c^2, d^2, e^2, f^2, g^2,\\
& (x,a), (x,g),\\
& (bc)^3, (bd)^2, (be)^2, (bf)^2, (bg)^2,\\
& (cd)^3, (ce)^2, (cf)^2, (cg)^2,\\
& (de)^3, (df)^2, (dg)^2,\\
& (ef)^3, (eg)^2,\\
& (fg)^3,\\
& (b, xbx), \\
& (a, edcb), (a,f)dcbdcd, (ag)^5,\\
& (cdef, xbx), (b,xcdefx), (cdef, xcdefx) >\end{aligned}
$$
we want to construct the coset table for the subgroup generated by $x, b, c, d, e, f, g$ interactively. First, we create a coset enumeration process. Since the index of the chosen subgroup is $1\,140\,000$, we request a coset limit of $1\,200\,000$. Then we start the enumeration.
```magma
> HN :=
> FPGroup< x, a, b, c, d, e, f, g |
> x^2, a^2, b^2, c^2, d^2, e^2, f^2, g^2,
> (x, a), (x, g),
> (b*c)^3, (b*d)^2, (b*e)^2, (b*f)^2, (b*g)^2,
> (c*d)^3, (c*e)^2, (c*f)^2, (c*g)^2,
> (d*e)^3, (d*f)^2, (d*g)^2,
> (e*f)^3, (e*g)^2,
> (f*g)^3,
> (b, x*b*x), (a, e*d*c*b), (a, f)*d*c*b*d*c*d,
> (a*g)^5, (c*d*e*f, x*b*x), (b, x*c*d*e*f*x),
> (c*d*e*f, x*c*d*e*f*x)
> >;
> H := sub;
> P := CosetEnumerationProcess(HN, H : CosetLimit := 1200000, Print := true);
> StartEnumeration(~P);
Overflow
(a=1110331 r=58605 h=101193 n=1200001;
l=5444 c=105.36;
m=1110331 t=2001537)
```
The enumeration could not be completed successfully. Though $P$ does contain a valid coset table, but this coset table fails to be closed.
```magma
> HasValidCosetTable(P);
true
> HasClosedCosetTable(P);
false
```
We extract the coset table map, check its domain and its codomain, and determine the position of the first “hole” in the coset table.
```magma
> ct := CosetTable(P);
> Domain(ct) : Minimal;
Cartesian Product<{ 1 .. 1110331 }, GrpFP: HN>
> Codomain(ct);
{ 0 .. 1110331 }
> row := 1;
> while forall(col){ gen : gen in {x, a, b, c, d, e, f, g}
> | ct() ne 0 } do
> row +:= 1;
> end while;
> row;
41881
> col;
x
```
We change the enumeration parameters for the process $P$, selecting the predefined strategy [`Hard`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess). Since this predefined strategy sets a workspace of $10\,000\,000$, we must expressly override this, if we want the old value to be retained. Because the workspace size never is decreased once an enumeration has been started, we can retain the old value simply by setting [`Workspace`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) to 0; this overrides the setting done by selecting the predefined strategy, but actually doesn’t change the process’ workspace. We then continue the enumeration with the new set of parameters, after checking that this is legal.
```magma
> SetProcessParameters(~P : Strategy := "Hard",
> Workspace := 0);
> CanContinueEnumeration(P);
true
> ContinueEnumeration(~P);
INDEX = 1140000
(a=1140000 r=58512 h=1144534 n=1144534;
l=70 c=6.50;
m=1140000 t=2035739)
```
We have obtained a closed coset table. Note that the codomain of the new coset table map does not contain 0.
```magma
> HasClosedCosetTable(P);
true
> ct := CosetTable(P);
> Domain(ct) : Minimal;
Cartesian Product<{ 1 .. 1140000 }, GrpFP: HN>
> Codomain(ct);
{ 1 .. 1140000 }
```
###### `Example: Copying a Coset Enumeration Process (ex-e1cc15)`
First, we create a coset enumeration process for the trivial subgroup of the finite group $G := \langle a, b\ | \ a^8, b^7, (a*b)^2, (a^{-1}*b)^3 \rangle$ and start the enumeration.
```magma
> F := FreeGroup(2);
> G := quo;
> H := sub;
> P := CosetEnumerationProcess(G, H : Print := true);
> StartEnumeration(~P);
INDEX = 10752
(a=10752 r=57263 h=1 n=57263;
l=292 c=0.11;
m=47825t=57262)
```
We want to enumerate the cosets of the two non-trivial subgroups of $G$ generated by $a^{-1}*b$ and $a^2$, respectively. To do this, we create two copies of the coset enumeration process `P` and use the function [`AddSubgroupGenerator`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-addsubgroupgenerator) for each of them. Since the copies inherit all information contained in `P`, we then can call the function [`RedoEnumeration`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-redoenumeration) to enumerate the cosets of the two non-trivial subgroups, making use of the existing coset table.
```magma
> P1 := P;
> AddSubgroupGenerator(~P1, a^-1*b);
> Subgroup(P1);
Finitely presented group on 2 generators
Generators as words in group G
$.1 = Id(G)
$.2 = a^-1 * b
> CanRedoEnumeration(P1);
true
> RedoEnumeration(~P1);
INDEX = 3584
(a=3584 r=57263 h=1 n=57263;
l=49 c=0.02;
m=47825 t=57262)
>
> P2 := P;
> AddSubgroupGenerator(~P2, a^2);
> Subgroup(P2);
Finitely presented group on 2 generators
Generators as words in group G
$.1 = Id(G)
$.2 = a^2
> CanRedoEnumeration(P2);
true
> RedoEnumeration(~P2);
INDEX = 2688
(a=2688 r=57263 h=1 n=57263;
l=37 c=0.02;
m=47825 t=57262)
```
Finally, we are interested in the quotient of $G$ by the normal closure of the subgroup generated by $a^4$ and want to enumerate the cosets of the image of the subgroup generated by $a^2$ in this quotient. Since this is the subgroup used in `P2`, we create a copy of `P2` and add the relation $a^4$. Again, we are able to make use of information obtained earlier, by continuing the inherited enumeration.
```magma
> P3 := P2;
> AddRelator(~P3, a^4);
> CanContinueEnumeration(P3);
true
> ContinueEnumeration(~P3);
INDEX = 84
(a=84 r=57263 h=1 n=57263;
l=2 c=0.00;
m=47825 t=57262)
```
We extract the quotient and its subgroup from the process `P3`, using the appropriate access functions.
```magma
> G3 := Group(P3);
> G3;
Finitely presented group G3 on 2 generators
Relations
a3^8 = Id(G3)
b3^7 = Id(G3)
(a3 * b3)^2 = Id(G3)
(a3^-1 * b3)^3 = Id(G3)
a3^4 = Id(G3)
> H3 := Subgroup(P3);
> H3;
Finitely presented group H3 on 2 generators
Generators as words in group G3
u3 = Id(G3)
v3 = a3^2
```
###### `Example: Normaliser Element from a Partial Coset Table (ex-ef1fe7)`
Consider the subgroup $H$ of the (infinite) group $G := \langle a, b\ | \ b^7, (a*b)^2, (a^{-1}*b)^3 \rangle$ generated by $a$. We create a coset enumeration process and start an enumeration with the default parameters.
```magma
> F := FreeGroup(2);
> G := quo;
> H := sub;
> P := CosetEnumerationProcess(G, H);
> StartEnumeration(~P : Print := true);
Overflow
(a=957026 r=415230 h=415230 n=999999;
l=3553 c=2.38;
m=960050 t=999998)
```
The enumeration produces a valid (albeit not complete) coset table.
```magma
> HasValidCosetTable(P);
true
> HasCompleteCosetTable(P);
false
```
Even the partial coset table is sufficient to find an element in the normaliser of $H$ in $G$.
```magma
> found, elt := ExistsNormalisingCoset(P);
> found;
true
> elt;
b^-4 * a * b^-2
```
###### `Example: Excluded Conjugates in a Coset Table (ex-0f03e7)`
Consider again the subgroup $H$ of the (infinite) group $G := \langle a, b\ | \ b^7, (a*b)^2, (a^{-1}*b)^3 \rangle$ generated by $a$. We create a coset enumeration process and start an enumeration with the default parameters.
```magma
> F := FreeGroup(2);
> G := quo;
> H := sub;
> P := CosetEnumerationProcess(G, H);
> StartEnumeration(~P : Print := true);
Overflow
(a=957026 r=415230 h=415230 n=999999;
l=3553 c=2.38;
m=960050 t=999998)
```
We check, whether the coset table exhibits excluded conjugates.
```magma
> ExistsExcludedConjugate(P);
true a^b
```
It does. This means in particular, that $H$ is not normal in $G$. We create a copy `P1` of the coset enumeration process `P`, extend the subgroup of `P1` by the excluded conjugates found in the previous step and restart the enumeration for `P1`.
```magma
> P1 := P;
> for c in ExcludedConjugates(P) do
> AddSubgroupGenerator(~P1, c);
> end for;
> RedoEnumeration(~P1);
INDEX = 1 (a=1 r=2 h=2 n=2; l=2 c=0.94; m=960050 t=999998)
```
The new subgroup is equal to $G$. In particular, the normal closure of $H$ in $G$ is the whole of $G$.
We return to the coset enumeration process `P` and check whether we can find a non-trivial element $x\in {\tt N}_G(H)$ such that $x^2 \in H$.
```magma
> ExistsCosetSatisfying(P : Order := 2, Normalizing := true);
true b^-4 * a * b^-2
```
We can. In fact, $b^{-4}\cdot a\cdot b^{-2}$ is the only non-trivial coset which is known to satisfy this condition…
```magma
> CosetsSatisfying(P : Order := 2, Normalizing := true);
{ Id(G), b^-4 * a * b^-2 }
```
…and we can’t find in a similar way a non-trivial element $x\in {\tt N}_G(H)$ such that $x^3 \in H$.
```magma
> ExistsCosetSatisfying(P : Order := 3, Normalizing := true);
false
```
Note the difference in the output of [`CosetSatisfying`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetsatisfying) and [`CosetsSatisfying`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying): The former takes into account only cosets other than $H$, whereas the latter (unless we set the parameter [`First`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetssatisfying)) includes the coset $H$, which obviously satisfies the specified conditions.
```magma
> CosetSatisfying(P : Order := 3, Normalizing := true);
{}
> CosetsSatisfying(P : Order := 3, Normalizing := true);
{ Id(G) }
```
#### Implicit Invocation of the Todd-Coxeter Algorithm
Several functions working with finitely presented groups at some point require a coset table of a subgroup and may invoke a coset enumeration indirectly, e.g. the function [`H meet K`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#operation-grpfp-1-meet) or the function [`Normaliser`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-normaliser). The default behaviour for such implicitly called coset enumerations is the same as the one for coset enumerations invoked explicitly, e.g. using the function [`ToddCoxeter`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-1-toddcoxeter).
If such an implicitly called coset enumeration fails to produce a closed coset table, the calling function may terminate with a runtime error.
Experienced users can control the behaviour of indirectly invoked coset enumerations with a set of global parameters. These global parameters are valid for all implicitly called coset enumerations. For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration). Note that coset enumerations which are *explicitly* invoked, e.g. by a call to the function [`Index`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-1-index), are not affected by this global set of parameters. Parameters for these functions have to be specified in the function call.
##### `SetGlobalTCParameters(: parameters)`
This function sets the parameter values used for indirect invocations of the Todd-Coxeter coset enumeration procedure. The parameters accepted and their default values are the same as for the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) described in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `UnsetGlobalTCParameters()`
This function restores the default values for the parameters used for indirect invocations of the Todd-Coxeter coset enumeration procedure. For a description of the meanings of the parameters and their default values, see [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Example: Implicit Coset Enumeration (ex-de08eb)`
We consider again the Harada-Norton simple group with the presentation
$$
\begin{aligned}< x, a, b, c, d, e, f, g\ | \ & x^2, a^2, b^2, c^2, d^2, e^2, f^2, g^2, (x,a), (x,g),\\
& (bc)^3, (bd)^2, (be)^2, (bf)^2, (bg)^2, (cd)^3, (ce)^2, (cf)^2, (cg)^2,\\
& (de)^3, (df)^2, (dg)^2, (ef)^3, (eg)^2, (fg)^3, (b, xbx),(a, edcb),\\
& (a,f)dcbdcd, (ag)^5, (cdef, xbx), (b,xcdefx), (cdef, xcdefx) >\end{aligned}
$$
and the subgroup $H$ generated by $x, b, c, d, e, f, g$.
```magma
> HN :=
> FPGroup< x, a, b, c, d, e, f, g |
> x^2, a^2, b^2, c^2, d^2, e^2, f^2, g^2,
> (x, a), (x, g),
> (b*c)^3, (b*d)^2, (b*e)^2, (b*f)^2, (b*g)^2,
> (c*d)^3, (c*e)^2, (c*f)^2, (c*g)^2,
> (d*e)^3, (d*f)^2, (d*g)^2,
> (e*f)^3, (e*g)^2,
> (f*g)^3,
> (b, x*b*x),
> (a, e*d*c*b), (a, f)*d*c*b*d*c*d, (a*g)^5,
> (c*d*e*f, x*b*x), (b, x*c*d*e*f*x),
> (c*d*e*f, x*c*d*e*f*x) >;
> H := sub;
```
$H$ has index 1,140,000 in $HN$. Using the default settings, the normaliser of $H$ in $HN$ cannot be computed.
```magma
> N := Normaliser(HN, H);
>> N := Normaliser(HN, H);
^
Runtime error in 'Normaliser': Coset table is not closed
```
We change the global parameters for implicitly called coset enumerations and try again.
```magma
> SetGlobalTCParameters( : Strategy := "Hard");
> N := Normaliser(HN, H);
```
With these parameters, the computation works. We see that $H$ is self-normalising in $HN$.
```magma
> Index(HN, N);
1140000
> IsSelfNormalising(HN, H);
true
```
### Coset Spaces and Tables
Let $G = < X | R >$ be a finitely presented group. Suppose that $H$ is a subgroup of $G$ having finite index $m$. Let $V = \lbrace c_1 (=H), c_2, \ldots, c_m \rbrace$ denote the set of distinct right cosets of $H$ in $G$. This set admits a natural $G$-action $f : V \times G \rightarrow V$ where
$$
f : < c_i, x > = c_k \quad\iff\quad c_i*x = c_k,
$$
for $c_i \in V$ and $x \in G$. The set $V$ together with this action is a $G$-set called a *right coset space* for $H$ in $G$. The action may also be represented using a coset table $T$. If certain of the products $c_i*x$ are unknown, the corresponding images under $f$ are undefined. In this case, $T$ is not closed, and $V$ is called an *incomplete coset space* for $H$ in $G$.
#### Coset Tables
A coset table is represented in Magma as a mapping. Given a finitely-presented group $G$ and a subgroup $H$, the corresponding (right) coset table is a mapping $f:\{1,\ldots,r\}\times G\rightarrow \{0,\ldots,r\}$, where $r$ is the index of $H$ in $G$. $f(i,x)$ is the coset to which coset $i$ is mapped under the action of $x\in G$. The value 0 is only included in the codomain if the coset table is not closed, and it denotes that the coset is not known.
##### `CosetTable(G, H: parameters): GrpFP, GrpFP -> Map`
The (right) coset table for $G$ over subgroup $H$, constructed by means of the Todd-Coxeter procedure. If the coset table does not close then the codomain will include the value 0.
Experienced users can control the Todd-Coxeter procedure invoked by this function with a wide range of parameters. This function accepts the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) described in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `CosetTableToRepresentation(G, T): GrpFP, Map -> Map, GrpPerm, Grp`
Given a coset table $T$ for a subgroup $H$ of $G$, construct the permutation representation of $G$ given by its action on the cosets of $H$, using the columns of $T$. The function returns:
**(a)**
The homomorphism $f: G \rightarrow P$;
**(b)**
The permutation group image $P$;
**(c)**
The kernel $K$ of the action (a subgroup of $G$).
##### `CosetTableToPermutationGroup(G, T): GrpFP, Map -> GrpPerm`
Given a coset table $T$ for a subgroup $H$ of $G$, construct the permutation group image of $G$ given by its action on the cosets of $H$, using the columns of $T$. This is the second return value of [`CosetTableToRepresentation`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-cosettabletorepresentation)`(G, T)`.
##### `Example: Coset Table in the Infinite Dihedral Group (ex-ae13a4)`
Consider the infinite dihedral group.
```magma
> G := DihedralFPGroup(0);
> G;
Finitely presented group G on 2 generators
Relations
b^2 = Id(G)
(a * b)^2 = Id(G)
```
We define a subgroup $S$ of $G$ and compute the coset table map for $S$ in $G$.
```magma
> S := sub;
> ct := CosetTable(G, S);
> ct;
Mapping from: Cartesian Product<{ 1 .. 10 }, GrpFP: G>
to { 1 .. 10 }
$1 $2 -$1
1. 2 2 3
2. 4 1 1
3. 1 4 5
4. 6 3 2
5. 3 6 7
6. 8 5 4
7. 5 8 9
8. 10 7 6
9. 7 10 10
10. 9 9 8
```
When printing the coset table, the action of the generators and of the non-trivial inverses of generators on the enumerated transversal is shown in table form.
Using the coset table, we now construct the permutation representation of $G$ on the cosets of $S$ in $G$. We assign the representation (a homomorphism), the image (a permutation group of degree $[G:S] = 10$) and the kernel of the permutation representation (a subgroup of $G$).
```magma
> fP, P, K := CosetTableToRepresentation(G, ct);
> fP;
Homomorphism of GrpFP: G into GrpPerm: P, Degree 10,
Order 2^2 * 5 induced by
a |--> (1, 2, 4, 6, 8, 10, 9, 7, 5, 3)
b |--> (1, 2)(3, 4)(5, 6)(7, 8)(9, 10)
```
Note that the images of $a$ and $b$ correspond to the first two columns of the printed coset table above.
```magma
> P;
Permutation group P acting on a set of cardinality 10
Order = 20 = 2^2 * 5
(1, 2, 4, 6, 8, 10, 9, 7, 5, 3)
(1, 2)(3, 4)(5, 6)(7, 8)(9, 10)
> K;
Finitely presented group K
Index in group G is 20 = 2^2 * 5
Subgroup of group G defined by coset table
```
Now, we define a subgroup of infinite index in $G$ and compute a coset table for it.
```magma
> H := sub;
> ct := CosetTable(G, H);
```
Of course, the coset table cannot be complete; note that 0 is in its codomain, indicating unknown images of cosets.
```magma
> ct;
Mapping from: Cartesian Product<{ 1 .. 1333331 }, GrpFP: G>
to { 0 .. 1333331 }
```
#### Coset Spaces: Induced Homomorphism
##### `CosetAction(G, H): Grp, Grp -> Hom(Grp), GrpPerm, Grp`
Given a subgroup $H$ of the group $G$, this function constructs the permutation representation $\phi$ of $G$ given by the action of $G$ on the cosets of $H$. It returns:
**(a)**
The homomorphism $\phi$;
**(b)**
The image group $\phi(G)$.
**(c)**
(if possible) the kernel of $\phi$.
The permutation representation is obtained by using the Todd-Coxeter procedure to construct the coset table for $H$ in $G$. Note that $G$ may be an infinite group: it is only necessary that the index of $H$ in $G$ be finite.
##### `CosetAction(V): GrpFPCos -> Hom(Grp), GrpPerm`
Construct the permutation representation $L$ of $G$ given by the action of $G$ on the coset space $V$. It returns the permutation representation $\phi$ (as a map) and its image.
##### `CosetImage(G, H): Grp, Grp -> GrpPerm`
Construct the image of $G$ given by its action on the (right) coset space of $H$ in $G$.
##### `CosetImage(V): GrpFPCos -> GrpPerm`
Construct the image of $G$ as defined by its action on the coset space $V$.
##### `CosetKernel(G, H): GrpFP, GrpFP -> GrpFP`
The kernel of $G$ in its action on the (right) coset space of $H$ in $G$. (Only available when the index of $H$ in $G$ is very small).
This function may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `CosetKernel(V): GrpFPCos -> GrpFP`
The kernel of $G$ in its action on the (right) coset space $V$. (Only available when the index of the subgroup $H$ of $G$ defining the coset space is very small).
##### `Example: Induced Homomorphism for Co1 (ex-a27ede)`
The first Conway group has a representation as the image of the group
$$
\begin{aligned}G =\ \end{aligned}
$$
under the homomorphism defined by the action of $G$ on the cosets of the subgroup $H =\ .$ The permutation group can be constructed as follows:
```magma
> F := FreeGroup(8);
> G := quo (s*t)^3, (s*u)^2, (s*v)^2, (s*w)^4, (s*x)^2, (s*y)^2, (s*z)^3,
> (t*u)^5, (t*v)^2, (t*w)^2, (t*x)^2, (t*y)^4, (t*z)^4,
> (u*v)^3, (u*w)^3, (u*x)^4, (u*y)^2, (u*z)^2,
> (v*w)^2, (v*x)^3, (v*y)^2, (v*z)^2,
> (w*x)^6, (w*y)^2, (w*z)^2,
> (x*y)^4, (x*z)^6,
> (y*z)^2,
> s*(u*x)^2 = (s*v*x*z)^3 = t*(w*x)^3 = (t*s*w*x*y)^3 = 1,
> (u*w*x)^7 = v*(t*z)^2 = v*(s*w*z)^3 = w*(t*y)^2 = 1>;
> H := sub< G | a, b, c, d, e, f, g, (a*d*e*f*c*e*f*g*h)^39 >;
> V := CosetSpace(G, H: FillFactor := 100000);
> Co1 := CosetImage(V);
> Degree(Co1);
98280
```
##### `Example: Permutation Representation of G2(3) (ex-6d4ee4)`
The group $G_2(3)$ is a homomorph of the fp-group $G$ defined below. We construct a permutation representation $G1$ for $G_2(3)$ on $351$ points, and then compute the subgroup generated by the images of the first four generators of $G$ in $G1$. (The functions applied to permutation groups are described in Chapter [Permutation Groups](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm).)
```magma
> F := FreeGroup(7);
> z := y*c*a^2*b;
> u := x*d;
> t := w*c*a*d*b^2*c;
> G, g :=
> quo< F | a^4, b^4, c^2, d^2, (a,b), (a*c)^2, (b*c)^2,
> (c*d)^2, d*a*d*b^-1, y^3, (a^-1*b)^y*d*a^-1*b^-1,
> (c*d*a^-1*b)^y*b^-1*a*d*c, a*d*y*d*a^-1*y, x^3,
> a^x*b^-1, b^x*a*b, c^x*c, (x*d)^2, (u*z)^6, w^3,
> (w,y), (a*b)^w*b^-1*a*d*c, (c*d*a^-1*b)^w*d*c*b^2,
> (b^2*c*d)^w*a^-1*b^-1, (c*a^2*b*w)^2,
> (a^-1*b)^w*d*a^-1*b^-1, (t*u)^3 >;
> z1 := g(z);
> u1 := g(u);
> t1 := g(t);
> H := sub< G | z1*a^2*b^2, u1*c, t1*a^2*b^2 >;
> f, G1, K := CosetAction(G, H);
> Degree(G1);
351
> print Order(G1), FactoredOrder(G1);
4245696 [ <2, 6>, <3, 6>, <7, 1>, <13, 1> ]
> CompositionFactors(G1);
G
| G(2, 3)
1
> S := sub< G1 | f(a), f(b), f(c), f(d) >;
> BSGS(S);
> S;
Permutation group S of degree 351
Order = 64 = 2^6
```
Thus, the images of a, b, c, and d in G1 generate the Sylow $2$-subgroup.
#### Coset Spaces: Construction
The indexed (right) coset space $V$ of the subgroup $H$ of the group $G$ is a $G$-set consisting of the set of integers $\{ 1, ..., m\}$, where $i$ represents the right coset $c_i$ of $H$ in $G$. The action of $G$ on this $G$-set is that induced by the natural $G$-action
$$
f : V \times G \rightarrow V
$$
where
$$
f : < c_i, x > = c_k \quad\iff\quad c_i*x = c_k,
$$
for $c_i \in V$ and $x \in G$. If certain of the products $c_i*x$ are unknown, the corresponding images under $f$ are undefined, and $V$ is called an *incomplete coset space* for $H$ in $G$.
##### `CosetSpace(G, H: parameters): GrpFP, GrpFP -> GrpFPCos`
This function attempts to construct a coset space for the subgroup $H$ in the group $G$ by means of the Todd-Coxeter procedure. If the enumeration fails to complete, the function returns an incomplete coset space. The coset space is returned as an indexed right coset space. For a sample application of this function see Example [Working with a Coset Space](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-886b74).
Experienced users can control the Todd-Coxeter procedure invoked by this function with a wide range of parameters. This function accepts the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) described in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `RightCosetSpace(G, H: parameters): GrpFP, GrpFP -> GrpFPCos`
##### `LeftCosetSpace(G, H: parameters): GrpFP, GrpFP -> GrpFPCos`
The explicit right coset space of the subgroup $H$ of the group $G$ is a $G$-set containing the set of right cosets of $H$ in $G$. The elements of this $G$-set are the pairs $< H, x >$, where $x$ runs through a transversal for $H$ in $G$. Similarly, the explicit left coset space of $H$ is a $G$-set containing the set of left cosets of $H$ in $G$, represented as the pairs $< x, H >$. These functions use the Todd-Coxeter procedure to construct the explicit right (left) coset space of the subgroup $H$ of the group $G$. For a sample application see Example [Working with a Coset Space](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#example-ex-886b74).
Experienced users can control the Todd-Coxeter procedure invoked by these functions with a wide range of parameters. Both functions accept the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) described in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
#### Coset Spaces: Elementary Operations
##### `H * g: GrpFP, GrpFPElt -> GrpFPCosElt`
Right coset of the subgroup $H$ of the group $G$, where $g$ is an element of $G$ (as an element of the right coset of $H$).
##### `C * g: GrpFPCosElt, GrpFPElt -> GrpFPCosElt`
Coset to which the right coset $C$ of the group $G$ is mapped by the (right) action of $g$, where $g$ is an element of $G$.
##### `C * D: GrpFPCosElt, GrpFPCosElt -> GrpFPCosElt`
Given two right cosets of the same normal subgroup $H$ of the group $G$, return the right coset that is the product of $C$ and $D$.
##### `g in C: GrpFPElt, GrpFPCosElt -> BoolElt`
Return `true` if element $g$ of group $G$ lies in the coset $C$.
##### `g notin C: GrpFPElt, GrpFPCosElt -> BoolElt`
Return `true` if element $g$ of group $G$ does not lie in the coset $C$.
##### `C1 eq C2: GrpFPCosElt, GrpFPCosElt -> BoolElt`
Returns `true` if the coset $C1$ is equal to the coset $C2$.
##### `C1 ne C2: GrpFPCosElt, GrpFPCosElt -> BoolElt`
Returns `true` if the coset $C1$ is not equal to the coset $C2$.
#### Accessing Information
##### `# V: GrpFPCos -> RngIntElt`
The cardinality of the coset space $V$.
##### `Action(V): GrpFPCos -> Map`
The mapping $V \times G \rightarrow V$ giving the action of $G$ on the coset space $V$. This mapping is a coset table.
##### ` @ T: GrpFPCosElt, GrpFPElt, Map -> GrpFPElt`
##### `T(i, w): Map, GrpFPCosElt, GrpFPElt -> GrpFPElt`
The image of coset $i$ as defined in the coset table $T$, under the action of word $w$.
##### `ExplicitCoset(V, i): GrpFPCos, RngIntElt -> GrpFPCosElt`
The element of the explicit coset space that corresponds to indexed coset $i$.
##### `IndexedCoset(V, w): GrpFPCos, GrpFPElt -> GrpFPCosElt`
The element of the indexed coset space $V$ to which the element $w$ of $G$ corresponds.
##### `IndexedCoset(V, C): GrpFPCos, GrpFPCosElt -> GrpFPCosElt`
The element of the indexed coset space $V$ to which the explicit coset $C$ of $G$ corresponds.
##### `Group(V): GrpFPCos -> GrpFP`
The group $G$ for which $V$ is a coset space.
##### `Subgroup(V): GrpFPCos -> GrpFP`
The subgroup $H$ of $G$ such that $V$ is a coset space for $G$ over $H$.
##### `IsComplete(V): GrpFPCos -> BoolElt`
Returns `true` if the coset space $V$ is complete.
##### `ExcludedConjugates(V): GrpFPCos -> { GrpFPElt }`
##### `ExcludedConjugates(T): Map -> { GrpFPElt }`
Given a partial or complete coset space $V$ for the group $G$ over the subgroup $H$, or a coset table $T$ corresponding to this coset space, this function returns the set of words $E =\{g_i^{-1}h_jg_i\ |\ g_i$ a generator of $G$, $h_j$ a generator of $H$, and, modulo $V$, $g_i^{-1}h_jg_i$ does not lie in $H\}$. If $E$ is empty, then $H$ is a normal subgroup of $G$, while if $E$ is non-empty, the addition of the elements of $E$ to the generators of $H$ will usually be a larger subgroup of the normal closure of $H$. This function may be used with incomplete coset spaces for $G$ over $H$; it may then happen that some of the elements of $E$ actually lie in $H$ but there is insufficient information for this to be detected. This function is normally used in conjunction with the Todd-Coxeter algorithm when seeking some subgroup having index sufficiently small so that the Todd-Coxeter procedure completes. The conjugates are returned as a set of words.
##### `Transversal(G, H): GrpFP, GrpFP -> {@ GrpFPElt @}, Map`
##### `RightTransversal(G, H): GrpFP, GrpFP -> {@ GrpFPElt @}, Map`
Given a finitely presented group $G$ and a subgroup $H$ of $G$, this function returns
**(a)**
A set of elements $T$ of $G$ forming a right transversal for $G$ over $H$; and
**(b)**
The corresponding transversal mapping $\phi: G \rightarrow T$. If $T = [t_1, \ldots, t_r]$ and $g \in G$, $\phi$ is defined by $\phi(g) = t_i$, where $g\in H*t_i$.
These functions may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Example: Accessing a Coset Table (ex-e36cb8)`
Consider the infinite dihedral group.
```magma
> G := DihedralFPGroup(0);
```
We define a subgroup $H$ of index 10 in $G$…
```magma
> H := sub< G | a*b, a^10 >;
> Index(G, H);
10
```
…and construct a right transversal for $H$ in $G$ and the associate transversal map.
```magma
> RT, transmap := Transversal(G, H);
> RT;
{@ Id(G), a, a^-1, a^2, a^-2, a^3, a^-3, a^4, a^-4, a^5 @}
> transmap;
Mapping from: GrpFP: G to SetIndx: RT
```
From this a left transversal is easily obtained:
```magma
> LT := {@ x^-1 : x in RT @};
> LT;
{@ Id(G), a^-1, a, a^-2, a^2, a^-3, a^3, a^-4, a^4, a^-5 @}
```
We construct the coset table and check whether the enumeration of the cosets is compatible to the enumeration of the right transversal $RT$.
```magma
> ct := CosetTable(G, H);
> forall(culprit){ i : i in [1..Index(G, H)]
> | ct(1, RT[i]) eq i};
true
```
It is. Thus, we can very easily define a function $RT\times G \rightarrow RT$, describing the action of $G$ on the set of right cosets of $H$ in $G$.
```magma
> action := func< r, g | RT[ct(Index(RT, r), g)] >;
```
Note that the definition of the function `action` relies on the fact that the computed right transversal and its enumeration match the ones internally used for the coset table $ct$.
```magma
> action(Id(G), b);
a
```
i.e. $H*b = H*a$.
```magma
> action(a^-4, a*b);
a^4
```
i.e. $Ha^{-4}*(ab) = Ha^4$.
##### `Example: Working with a Coset Space (ex-886b74)`
Consider the group $G =< a, b\ |\ a^8, b^7, (ab)^2, (a^{-1}b)^3 >$ and the subgroup $H$ of $G$, generated by $a^2$ and $a^{-1}b$.
```magma
> F := FreeGroup(2);
> G := quo< F | x^8, y^7, (x*y)^2, (x^-1*y)^3 >;
> H := sub< G | a^2,a^-1*b >;
```
We construct an indexed right coset space $V$ and an explicit right coset space $V\!\!r$ for $H$ in $G$.
```magma
> V := CosetSpace(G, H);
> Vr := RightCosetSpace(G, H);
```
The coset $H$ always has index 1.
```magma
> trivial_coset := ExplicitCoset(Vr, 1);
> trivial_coset;
> IndexedCoset(V, trivial_coset);
1
```
We now pick a coset…
```magma
> coset := ExplicitCoset(Vr, 42);
> coset;
```
…multiply from the right with $b$…
```magma
> coset * b;
```
…and check where this gets us in the indexed coset space $V$.
```magma
> IndexedCoset(V, coset * b);
23
```
##### `Example: Derived Subgroup via Coset Enumeration (ex-13317b)`
We present a function which computes the derived subgroup $G'$ for the finitely presented group $G$. It assumes that the Todd-Coxeter procedure can enumerate the coset space of $G'$ in $G$.
```magma
> function DerSub(G)
>
> /* Initially define S to contain the commutator of each pair of distinct
> generators of G */
>
> S := { (x,y) : x, y in Generators(G) | (x,y) ne Id(G) };
>
> /* successively extend S until it is closed under conjugation by the
> generators of G */
>
> repeat
> V := CosetSpace(G, sub);
> E := ExcludedConjugates(V);
> S := S join E;
> until # E eq 0;
> return sub;
> end function;
```
##### `Example: Adding Excluded Conjugates (ex-8d212e)`
Given a subgroup $H$ of the finitely presented group $G$, for which the Todd-Coxeter procedure does not complete, add excluded conjugates one at a time to the generators of $G$ until a subgroup $K$ is reached such that either $K$ is normal in $G$, or $K$ has sufficiently small index for the Todd-Coxeter method to complete. The set $hgens$ contains a set of generating words for $H$.
```magma
> function NormClosure(G, hgens)
> xgens := hgens;
> kgens := hgens;
> indx := 0;
> while # xgens ne 0 do
> Include(~kgens, Representative(xgens));
> V := CosetSpace(G, sub);
> if IsComplete(V) then break; end if;
> xgens := ExcludedConjugates(V);
> end while;
> if IsComplete(V) then
> print "The subgroup generated by", kgens, "has index", #V;
> return kgens;
> else
> print "The construction was unsuccessful";
> return {};
> end if;
> end function;
```
#### Double Coset Spaces: Construction
##### `DoubleCoset(G, H, g, K): GrpFP, GrpFP, GrpFPElt, GrpFP -> GrpFPDcosElt`
The double coset $H*g*K$ of the subgroups $H$ and $K$ of the group $G$, where $g$ is an element of $G$.
##### `DoubleCosets(G, H, K): GrpFP, GrpFP, GrpFP -> { GrpFPDcosElt }`
Set of double cosets $H*g*K$ of the group $G$.
This function may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Example: Double Cosets in the Infinite Dihedral Group (ex-b7c1be)`
Consider again the infinite dihedral group $G$…
```magma
> G := DihedralFPGroup(0);
```
…and the subgroup $H$ of index 10 in $G$.
```magma
> H := sub< G | a*b, a^10 >;
```
The set of $H$-$H$ double cosets in $G$ can be obtained with the statement
```magma
> DoubleCosets(G, H, H);
{ , ,
, ,
, }
```
#### Coset Spaces: Selection of Cosets
##### `CosetsSatisfying(T, S: parameters): Map, { GrpFPElt } -> { GrpFPCosElt }`
##### `CosetSatisfying(T, S: parameters): Map, { GrpFPElt } -> { GrpFPCosElt }`
##### `CosetsSatisfying(V, S: parameters): GrpFPCos, { GrpFPElt } -> { GrpFPCosElt }`
##### `CosetSatisfying(V, S: parameters): GrpFPCos, { GrpFPElt } -> { GrpFPCosElt }`
```magma
First : RngIntElt Default: 1
Last : RngIntElt Default: ＃V
Limit : RngIntElt Default: Infinity()
Normalizing: BoolElt Default: false
Order : RngIntElt Default: 0
Print : RngIntElt Default: 0
```
Given a fp-group $G$, and a partial or complete coset space $V$ or coset table $T$ for $G$ over the subgroup $H$ generated by the set of words $S$, these functions return representatives for the cosets of $V$ which satisfy the conditions defined in the parameters. In the description of the parameters below, the symbol $l$ will denote a Boolean value, while the symbol $n$ will denote a positive integer in the range [1, #$V$].
The functions are not identical. `CosetsSatisfying` returns a set of coset representatives for $V$ as defined in the parameters. `CosetSatisfying` is the same as `CosetsSatisfying` with the `Limit` parameter equal to 1; thus it returns a set containing a single coset representative, or an empty set if no cosets satisfy the conditions.
Start looking for coset representatives satisfying the designated conditions beginning with coset $i$ of $V$.
Stop looking for coset representatives after examining coset $j$ of $V$.
Terminate the search for coset representatives as soon as $l$ have been found which satisfy the designated conditions. This parameter is not available for `CosetSatisfying`, since `Limit` is set to 1 for this function.
If `true`, select coset representatives $x$ such that, modulo $V$, the word $x^{-1}h_1x$, $\ldots$, $x_1h_sx$ is contained in $H$.
Select coset representatives $x$ such that, modulo $V$, the word $x^n$ is contained in $H$.
If $t > 0$, print the coset representatives found to satisfy the designated conditions.
##### `Example: Selecting Cosets by a Condition (ex-c086e8)`
Consider the braid group $G$ on 4 strings with Artin generators $a$, $b$ and $c$ and the subgroup $H$ of $G$ generated by $a$ and $b$.
```magma
> G := BraidFPGroup(4);
> H := sub< G | a,b >;
```
We construct an – obviously incomplete – explicit right coset space for $H$ in $G$.
```magma
> V := RightCosetSpace(G, H);
```
Using the function [`CosetSatisfying`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-cosetsatisfying), we compute an element of $G$, not contained in $H$, which normalises $H$.
```magma
> cs := CosetSatisfying(V, Generators(Subgroup(V))
> : Normalizing := true, First := 2);
> cs;
{
}
> rep := c * b * a^2 * b * c;
```
The conjugates of $a$ and $b$ by this element had better be in $H$…
```magma
> rep^-1 * a * rep in H;
true
> rep^-1 * b * rep in H;
true
```
…OK.
#### Constructing a Presentation for a Subgroup
Let $H$ be a subgroup of finite index in the finitely presented group $G$. It frequently happens that it is desirable to construct a set of defining relations for $H$ from those of $G$. Such a presentation can be obtained either on a set of Schreier generators for $H$ or on the given generators of $H$ using the Reidemeister-Schreier rewriting technique [[MKS76](https://docs.magma-maths.org/references.md#cite-magnus-cgt)], if necessary together with extended coset enumeration [[AR84](https://docs.magma-maths.org/references.md#cite-extended-tc), [HKRR84](https://docs.magma-maths.org/references.md#cite-havas-tietze)]. We emphasise that if the user wishes only to determine the structure of the maximal abelian quotient of $H$, then the function [`AbelianQuotientInvariants`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotientinvariants) should be used. In this case there is no need to first construct a presentation for $H$ using the [`Rewrite`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#function-grpfp-1-rewrite) function described below, since [`AbelianQuotientInvariants`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotientinvariants) employs a special form of the Reidemeister-Schreier rewriting process which abelianises each relator as soon as it is constructed. Thus, compared to the function `Rewrite`, the function `AbelianQuotientInvariants` can be applied to subgroups of much larger index.
##### `Rewrite(G, H : parameters): GrpFP, GrpFP -> GrpFP, Map`
```magma
Simplify : BoolElt Default: true
EliminationLimit : RngIntElt Default: 100
ExpandLimit : RngIntElt Default: 150
GeneratorsLimit : RngIntElt Default: 0
LengthLimit : RngIntElt Default: Infinity()
SaveLimit : RngIntElt Default: 10
SearchSimultaneous: RngIntElt Default: 20
Iterations : RngIntElt Default: 10000
Print : RngIntElt Default: 0
```
Given a finitely presented group $G$ and a subgroup $H$ having finite index in $G$, return a group $R$ isomorphic to $H$ with a presentation on (some of) the Schreier generators of $H$ in $G$. The group $R$ will be created as a subgroup of $G$ and defining relations of $R$ on its generators will be available. Note that the generators of $R$ will, in general, not correspond to the generators of $H$. The isomorphism from $H$ onto $R$ is returned as second return value.
This function may require the computation of a coset table. Experienced users can control the behaviour of a possibly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
If this Boolean-valued parameter is given the value `true`, then the resulting presentation for $H$ will be simplified (default). The function `Rewrite` returns a finitely presented group that is isomorphic to $H$. If simplification is requested (by setting `Simplify := true`) then the simplification procedures are invoked (see next section). These procedures perform a sequence of Tietze transformations which typically result in a considerable simplification of the presentation produced by the rewriting process. Alternatively, the user can set `Simplify := false` and then perform the simplification directly if desired. (See next section). If simplification is not requested as part of `Rewrite`, a small amount of simplification is performed on the presentation before it is returned.
These parameters control the simplification. See the description of [`Simplify`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/simplification.md#function-grpfp-1-simplify) for an explanation of these parameters.
##### `Rewrite(G, ~H : parameters): GrpFP, GrpFP`
```magma
Simplify: BoolElt Default: true
```
Given a finitely presented group $G$ and a subgroup $H$ having finite index in $G$, compute a defining set of relations for $H$ on the existing generators, using extended coset enumeration and Reidemeister-Schreier rewriting, and change the presentation of $H$ accordingly.
If the computation is successful, defining relations for $H$ on its generators will be available at the end; any previously computed relations of $H$ will be discarded. If the computation is unsuccessful, $H$ is not changed. In any case, both the isomorphism type of $H$ and its embedding into $G$ as a subgroup is preserved by this function.
If this parameter is given the value `true` (default), then an attempt will be made to simplify the constructed set of relations by substring searches, that is, Tietze transformations not changing any generators. The generating set of $H$ is not modified by this process.
Moreover, the extended coset enumeration can be controlled by a wide range of parameters. The function `Rewrite` – in addition to the parameter `Simplify` – accepts the same parameters as the function [`CosetEnumerationProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#function-grpfp-cosetenumerationprocess) described in Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `Example: Presentation for an Index 3 Subgroup (ex-e234d8)`
Starting with the group $G$ defined by
$$
< x, y\ |\ x^2, y^3, (xy)^{12}, (xy)^6(xy^{-1})^6 >,
$$
we construct a subgroup $K$ of index $3$ generated by the words $x$, $yxy^{-1}$, and $yxy^{-1}xy^{-1}xy$. We present the subgroup $K$, compute its abelian quotient structure, and then show that the class $30$ 2-quotient of $K$ has order $2^{62}$.
```magma
> G := FPGroup< x, y | x^2, y^3, (x*y)^12, (x*y)^6*(x*y^-1)^6 >;
> G;
Finitely presented group G on 2 generators
Relations
x^2 = Id(G)
y^3 = Id(G)
(x * y)^12 = Id(G)
x * y * x * y * x * y * x * y * x * y * x * y * x * y^-1 * x *
y^-1 * x * y^-1 * x * y^-1 * x * y^-1 * x * y^-1 = Id(G)
> K := sub< G | x, y*x*y^-1, y*x*y^-1*x*y^-1*x*y >;
> K;
Finitely presented group K on 3 generators
Generators as words in group G
K.1 = x
K.2 = y * x * y^-1
K.3 = y * x * y^-1 * x * y^-1 * x * y
> Index(G, K);
3
> T := Rewrite(G, K);
> T;
Finitely presented group T on 3 generators
Generators as words in group G
T.1 = x
T.2 = y * x * y^-1
T.3 = x^y
Relations
T.1^2 = Id(T)
T.2^2 = Id(T)
T.3^2 = Id(T)
(T.3 * T.2 * T.1 * T.3 * T.2)^2 = Id(T)
(T.1 * T.3 * T.2 * T.1 * T.3)^2 = Id(T)
(T.1 * T.2 * T.1 * T.3 * T.2)^2 = Id(T)
> AbelianQuotientInvariants(T);
[ 2, 2, 2 ]
> Q2 := pQuotient(T, 2, 30);
> FactoredOrder(Q2);
[ <2, 62> ]
```
##### `Example: Presentation on New Generators (ex-490fe5)`
In this example we illustrate how the function [`Rewrite`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#procedure-grpfp-1-rewrite-var) can be used to obtain a presentation of a finitely presented group on a different set of generators.
We start with a presentation of $L_2(7)$ on two generators $x$ and $y$.
```magma
> F := FPGroup< x, y | x^3 = 1, y^3 = 1, (x*y)^4 = 1,
> (y*y^x)^2 = y^x*y >;
```
The group is also generated by the elements $a = (xy)^2$ and $b = y$.
```magma
> H := sub;
> Index(F,H);
1
```
At the moment, no defining relations of $H\cong F$ on the generators $a$ and $b$ are known.
```magma
> H;
Finitely presented group H on 2 generators
Index in group F is 1
Generators as words in group F
a = (x * y)^2
b = y
```
We apply the function [`Rewrite`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/coset-space-table.md#procedure-grpfp-1-rewrite-var) to $H$ as a subgroup of $F$ in order to compute defining relations on the generators $a$ and $b$.
```magma
> Rewrite(F, ~H);
> H;
Finitely presented group H on 2 generators
Index in group F is 1
Generators as words in group F
a = (x * y)^2
b = y
Relations
a^2 = Id(H)
b^3 = Id(H)
(a * b)^7 = Id(H)
(a * b^-1 * a * b)^4 = Id(H)
(b * a * b^-1 * a * b * a)^4 = Id(H)
```
The last relation turns out to be redundant; $a$ and $b$ are standard generators for $L_2(7)$.
```magma
> Order(DeleteRelation(H,5)) eq Order(H);
true
```
### Subgroups of Finite Index
The functions in this section are concerned with the construction of subgroups of finite index. We first describe a method for computing all subgroups whose index does not exceed some (modest) integer bound. The next family of functions are concerned with constructing new subgroups of finite index from one or more known ones.
#### Low Index Subgroups
If an fp-group $G$ has subgroups of (small) finite index then there algorithms which can be used to find such subgroups and then use them to determine some properties of $G$.
##### `LowIndexSubgroups(G, R : parameters): GrpFP, RngIntElt -> [ GrpFP ]`
##### `LowIndexSubgroups(G, R : parameters): GrpFP, RngIntElt -> [ { GrpFPElt } ]`
##### `LowIndexSubgroups(G, R: parameters): GrpFP, -> [ GrpFP ]`
##### `LowIndexSubgroups(G, R: parameters): GrpFP, -> [ { GrpFPElt } ]`
Given a finitely presented group $G$ (possibly the free group), and an expression $R$ defining a positive integer range (see below), determine the conjugacy classes of subgroups of $G$ whose indices lie in the range specified by $R$. The subgroups are generated by systematically building all coset tables consistent with the defining relations for $G$ and which satisfy the range condition $R$. The argument $R$ is one of the following:
**(a)**
An integer $n$ representing the range $[1, n]$;
**(b)**
A tuple $$ representing the range $[a, b]$;
The generation of subgroups can be controlled by a set of parameters described below. The returned sequence contains the subgroups found and is sorted in order of increasing index in $G$.
##### `Example: Tutte's 8-Cage and the Conder Graph (ex-2e2f0b)`
(Peter Lorimer) The two graphs known as Tutte’s 8-cage and the Conder graph may be constructed as the Cayley graphs of two conjugacy classes of subgroups having index $10$ in the finitely presented group
$$
\begin{aligned}< q, r, s, h, a \ |\ &h^3 = a^2 = p^2 = 1, p^h = p, p^a = q, q^h = r, r^a = s, \\
& h^s = h^{-1}, r^h = pqr, sr = pqrs, pq = qp, pr = rp, ps = sp, qr = rq, qs = sq>.\end{aligned}
$$
We use the low index function to construct these subgroups.
```magma
> G := FPGroup h^3 = a^2 = p^2 = 1, p^h = p, p^a = q,
> q^h = r, r^a = s, h^s = h^-1, r^h = p * q * r,
> s * r = p * q * r * s, p * q = q * p,
> p * r = r * p, p * s = s * p, q * r = r * q,
> q * s = s * q>;
> LowIndexSubgroups(G, <10, 10>);
[
Finitely presented group on 6 generators
Index in group G is 10 = 2 * 5
Generators as words in group G
$.1 = p
$.2 = s
$.3 = h
$.4 = q^-2
$.5 = a * h^-1 * a * r^-1
$.6 = a * h * a * h^-1 * a * q^-1,
Finitely presented group on 6 generators
Index in group G is 10 = 2 * 5
Generators as words in group G
$.1 = p
$.2 = s
$.3 = h
$.4 = q^-2
$.5 = a * h^-1 * a * r^-1
$.6 = a * h * a * h^-1 * a
]
```
##### `Example: Conjugacy Classes via Low Index Subgroups (ex-18108c)`
A fairly surprising application for the low index subgroup algorithm is the enumeration of the conjugacy classes of a finite fp-group. In this example, we consider the group
$$
G \simeq {{\operatorname{PGL}}}_2(9) =
\langle a,b | a^2, b^3, (ab)^8, [a,b]^5, [a,(ba)^3b^{-1}]^2 \rangle.
$$
```magma
> G := FPGroup< a,b | a^2, b^3, (a*b)^8, (a,b)^5,
> (a,(b*a)^3*b^-1)^2 >;
> Order(G);
720
```
In an infinite fp-group, finding all classes of subgroups up to an index of 720 by applying the low index subgroup algorithm, would be extremely hard. In the case of the finite group $G$, however, we succeed.
```magma
> time sgG := LowIndexSubgroups(G, Order(G));
Time: 31.859
> #sgG;
26
```
We get a list of 26 representatives of the conjugacy classes of subgroups. For every representative, its index in $G$ and a set of generating words are known. We just have a look at two of them.
```magma
> sgG[10];
Finitely presented group on 2 generators
Index in group G is 60 = 2^2 * 3 * 5
Generators as words in group G
$.1 = b
$.2 = a * b * a * b * a * b^-1 * a * b * a * b * a * b^-1 * a
* b^-1 * a
> sgG[21];
Finitely presented group on 1 generator
Index in group G is 180 = 2^2 * 3^2 * 5
Generators as words in group G
$.1 = (b * a)^2
```
The function [`LowIndexSubgroups`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-lowindexsubgroups) constructs all conjugacy classes of subgroups of $G$ satisfying the following two conditions:
**(i)**
The index of each subgroup is in the range defined by $R$;
**(ii)**
If the parameter `Subgroup` defines a subgroup $H$, then at least one subgroup in each conjugacy class contains the subgroup $H$.
The subgroups are returned as a sequence of subgroups $G$, unless otherwise specified by the parameter `GeneratingSets` (see below). The sequence is sorted by increasing index of the subgroups in $G$. The subgroups are constructed using an algorithm due to Sims [[Sim94](https://docs.magma-maths.org/references.md#cite-sims-fpgroups), sect. 5.6]. This algorithm constructs the coset tables by using a backtrack algorithm. At a given position in the coset table, coset definitions are made systematically. Once a new definition has been made, the group relations are traced in an attempt to deduce further entries or to infer that this partial table will not extend to a table corresponding to a new class of subgroups. When either it cannot define a new entry, or when a complete table has been constructed, the algorithm backtracks to try the next possibility (this may introduce a new row, increasing the index). This algorithm may also be run as a process in such a way that the subgroups are returned one at a time, thereby allowing the user to analyze each subgroup as soon as it is found.
```magma
ColumnMajor: BoolElt Default: false
```
If `ColumnMajor` is set `false` (*default*), then the location for a new definition in the coset table is determined by searching the table in row major order for undefined entries. If `ColumnMajor` is set `true`, then the position for a new definition is determined by searching the table in column major order. If the presentation for $G$ contains explicit relators expressing the fact that certain of the generators have large order, then the presentation should be organized so that these generators appear first and the column major order should be selected for new coset definition. This strategy often leads to greatly improved performance.
```magma
GeneratingSets: BoolElt Default: false
```
The conjugacy classes of subgroups are returned in the form of a sequence of sets of words, where the $i$-th set is a generating set for a representative subgroup from the $i$-th conjugacy class of subgroups satisfying the given conditions. This is a much more compact representation than returning the subgroups as a sequence of actual *subgroups* of $G$ and should be used when a very large number of subgroups is expected, as there may be insufficient space to store each of them as a subgroup.
```magma
Limit: RngIntElt Default: Infinity()
```
Terminate after finding $n$ conjugacy classes of subgroups satisfying the designated conditions.
```magma
Long: [ RngIntElt ] Default: []
```
This option enables the user to designate certain of the defining relators for $G$ as *long relators*. The relators of $G$ are numbered from $1$ to $r$, in the order they appear in the [`quo< F | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-quo)- or [`FPGroup< X | R >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/fp-group-construction.md#constructor-grpfp-1-confpgroup)-constructors. The value $L$ of `Long` is a subset of the integer set $\{ 1,\ldots, r\}$. Magma interprets the relators whose numbers appear in $L$ as long relators. A relator designated as long is not used during the construction of a coset table. Rather, it is applied once a complete table has been found. There is some evidence to suggest that better performance is achieved in those groups having one or more very long relators by deferring application of these relators until such time as a complete coset table has been obtained.
```magma
Print: RngIntElt Default: 0
```
A description of each class of subgroups may be printed immediately after it is constructed. The value $n$ assigned to the `Print` parameter specifies just what information is to be printed, according to the following rules:
**$n = 0$**
No printing (*default*).
**$n = 1$**
For each class, print a heading and a set of generators for the class representative.
**$n = 2$**
The information printed for $n = 1$, together with the permutation representation of $G$ on the right cosets of the class representative.
**$n = 3$**
The information printed for $n = 2$, together with generators for the normalizer $N$ of the class representative, and a system of right coset representatives for $N$ in $G$.
```magma
Subgroup: GrpFP Default: sub< G | >
```
By specifying a value $H$ for `Subgroup`, only subgroups containing $H$ will be constructed.
```magma
TimeLimit: RngIntElt Default: 0 (no limit)
```
A time limit in seconds. A value of 0 (default) means no limit.
##### `LowIndexProcess(G, R : parameters): GrpFP, RngIntElt -> Process(Lix)`
##### `LowIndexProcess(G, R: parameters): GrpFP, < RngIntElt, RngIntElt > -> Process(Lix)`
```magma
ColumnMajor : BoolElt Default: false
GeneratingSets: BoolElt Default: false
Long : [ RngIntElt ] Default: []
Print : RngIntElt Default: 0
Subgroup : GrpFP Default: sub< G | >
TimeLimit : RngIntElt Default: 0 (no limit)
```
Create a low index subgroups *process*. This process may be used to create the conjugacy classes of proper subgroups one at time, with control being handed back to the Magma language processor each time a new class of subgroups is found. This function returns a process which is used by the function [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup) to actually produce the subgroups. The arguments and parameters have the same interpretation as for the function [`LowIndexSubgroups`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-lowindexsubgroups), except that `Limit` is not available (since the same effect can be achieved by limiting the number of calls to [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup)). Setting a time limit for a process $P$ limits the total amount of time spent in calls to [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup). If the time limit is exceeded in a call to [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup), this call is aborted and $P$ becomes invalid. Further attempts to access $P$ will cause a runtime error. The function [`IsValid`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-isvalid-lix) can be used to check whether a process is valid in order to avoid runtime errors in loops or user written functions.
##### `NextSubgroup(~P): GrpFPLixProc`
##### `NextSubgroup(~P, ~G): GrpFPLixProc, GrpFP`
Given a low index subgroups process $P$, construct the next conjugacy class of proper subgroups. The process $P$ must have been previously created using the function [`LowIndexProcess`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-lowindexprocess) and must be valid. Calling `NextSubgroup` for an empty process has no effect.
##### `ExtractGroup(P): GrpFPLixProc -> GrpFP`
Extract a representative subgroup for the conjugacy class currently defined by the low index process $P$. The subgroup extracted will be the one found by the previous invocation of [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup), or the first subgroup if [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup) has never been invoked on this process. Note that `ExtractGroup` will not search for a new subgroup. If $P$ is empty or invalid, a runtime error will result.
##### `ExtractGenerators(P): GrpFPLixProc -> { GrpFPElt }`
Extract a generating set for the representative subgroup of the conjugacy class currently defined by the low index process $P$. The subgroup extracted will be the one found by the previous invocation of [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup), or the first subgroup if [`NextSubgroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#procedure-grpfp-1-nextsubgroup) has never been invoked on this process. Note that `ExtractGenerators` will not search for a new subgroup. If $P$ is empty or invalid, a runtime error will result.
##### `IsEmpty(P): GrpFPLixProc -> BoolElt`
Return `true` if the low index process $P$ has already found all conjugacy classes of subgroups. If `IsEmpty` is called immediately following the creation of the low index process, then it will return the value `false` if there are no subgroups satisfying the specified conditions or advance $P$ to the first such subgroup otherwise.
##### `IsValid(P): GrpFPLixProc -> BoolElt`
Return `true` if the low index process $P$ is valid, that is, no limit has been exceeded. If `IsValid` is called immediately following the creation of the low index process, then it will return the value `false` if no subgroups satisfying the specified conditions can be found within the specified time or advance $P$ to the first such subgroup otherwise.
##### `Example: Low Index Subgroups of a Triangle Group (ex-fd99df)`
We determine all conjugacy classes of subgroups having index at most $15$ in the triangle group
$$
.
$$
```magma
> G := FPGroup< a, b | a^2, b^3, (a*b)^7 >;
> L := LowIndexSubgroups(G, 15: Print := 1);
Subgroup class 1 Index 7 Length 7 Subgroup generators :-
{ a, b * a * b^-1, b^-1 * a * b * a * b^-1 * a * b }
Subgroup class 2 Index 7 Length 7 Subgroup generators :-
{ a, b^-1 * a * b^-1 * a * b * a * b, b * a * b^-1 }
Subgroup class 3 Index 15 Length 15 Subgroup generators :-
{ a, b^-1 * a * b * a * b^-1 * a * b * a * b^-1 * a * b, b * a * b^-1 }
Subgroup class 4 Index 15 Length 15 Subgroup generators :-
{ a, b * a * b^-1, b^-1 * a * b^-1 * a * b * a * b^-1 * a * b * a * b }
Subgroup class 5 Index 14 Length 14 Subgroup generators :-
{ a, b^-1 * a * b * a * b^-1 * a * b * a * b * a * b^-1 * a * b, b * a * b^-1 }
Subgroup class 6 Index 14 Length 7 Subgroup generators :-
{ a, b^-1 * a * b * a * b^-1 * a * b * a * b^-1 * a * b * a * b^-1 * a * b, b *
a * b * a * b^-1 }
Subgroup class 7 Index 14 Length 14 Subgroup generators :-
{ a, b^-1 * a * b * a * b^-1 * a * b^-1 * a * b * a * b * a * b^-1 * a * b, b *
a * b * a * b^-1 }
Subgroup class 8 Index 14 Length 7 Subgroup generators :-
{ a, b * a * b * a * b^-1 * a * b^-1 * a * b * a * b * a * b^-1 * a * b^-1, b^-1
* a * b * a * b }
Subgroup class 9 Index 15 Length 15 Subgroup generators :-
{ a, b * a * b * a * b^-1 * a * b^-1, b^-1 * a * b^-1 * a * b * a * b }
Subgroup class 10 Index 9 Length 9 Subgroup generators :-
{ a, b * a * b * a * b * a * b^-1 }
Subgroup class 11 Index 14 Length 7 Subgroup generators :-
{ a, b * a * b * a * b * a * b^-1 * a * b, b * a * b^-1 * a * b * a * b^-1 }
Subgroup class 12 Index 14 Length 7 Subgroup generators :-
{ a, b * a * b * a * b^-1 * a * b * a * b, b * a * b^-1 * a * b * a * b^-1 }
Subgroup class 13 Index 14 Length 7 Subgroup generators :-
{ a, b^-1 * a * b * a * b^-1 * a * b, b * a * b * a * b * a * b^-1 * a * b^-1 }
Subgroup class 14 Index 14 Length 7 Subgroup generators :-
{ a, b * a * b * a * b^-1 * a * b * a * b, b^-1 * a * b * a * b^-1 * a * b }
Subgroup class 15 Index 14 Length 14 Subgroup generators :-
{ a, b^-1 * a * b * a * b * a * b^-1 * a * b, b * a * b * a * b * a * b^-1 * a *
b^-1 }
Subgroup class 16 Index 8 Length 8 Subgroup generators :-
{ b, a * b * a * b * a * b^-1 * a }
```
##### `Example: PSL(2,8) as a Quotient of a Triangle Group (ex-55eb8b)`
In this example we illustrate the use of the low index subgroup process by using it to determine whether the simple group ${\operatorname{PSL}}(2, 8)$ is a homomorphic image of the triangle group
$$
< x, y\ |\ x^2, y^3, (xy)^7 >.
$$
```magma
> F := FreeGroup(2);
> G := quo< F | x^2, y^3, (x*y)^7 >;
> LP := LowIndexProcess(G, 30);
> i := 0;
> while i le 100 and not IsEmpty(LP) do
> H := ExtractGroup(LP);
> NextSubgroup(~LP);
> P := CosetImage(G, H);
> if Order(P) eq 504 and IsSimple(P) then
> print "The group G has L(2, 8) as a homomorphic image.";
> print "It is afforded by the subgroup:-", H;
> break;
> end if;
> i +:= 1;
> end while;
The group G has L(2, 8) as a homomorphic image.
It is afforded by the subgroup:-
Finitely presented group H on 4 generators
Index in group G is 28 = 2^2 * 7
Generators as words in group G
H.1 = x
H.2 = y * x * y^-1
H.3 = y^-1 * x * y * x * y^-1 * x * y * x * y^-1 * x * y * x * y^-1 *
x * y * x * y
H.4 = y^-1 * x * y * x * y^-1 * x * y^-1 * x * y * x * y^-1 * x * y *
x * y * x * y^-1 * x * y
```
##### `Example: Proving a Group Is Infinite (ex-dd899e)`
This example shows how the low index subgroup machinery may be used as part of a function trying to prove that a group is infinite:
```magma
> function MyIsInfinite(G)
>
> // ...
>
> // Low index subgroup approach: check whether an obviously
> // infinite subgroup can be found in reasonable time.
> P := LowIndexProcess(G, 30 : TimeLimit := 5);
> while IsValid(P) and not IsEmpty(P) do
> H := ExtractGroup(P);
> NextSubgroup(~P);
> if 0 in AbelianQuotientInvariants(H) then
> print "The group G has subgroup:-", H;
> print "whose abelian quotient is infinite";
> print "Hence G is infinite.";
> return true;
> end if;
> end while;
> print "Low index approach fails; trying other methods...";
>
> // ...
>
> end function;
```
We try the code fragment on the group
$$
< x, z\ |\ z^3 x z^3 x^{-1}, z^5 x^2 z^2 x^2 >.
$$
```magma
> G := FPGroup;
> MyIsInfinite(G);
The group G has subgroup:-
Finitely presented group H on 4 generators
Index in group G is 4 = 2^2
Generators as words in group G
H.1 = x
H.2 = z * x * z
H.3 = z^3
H.4 = z * x^-1 * z * x * z^-1
whose abelian quotient has structure [ 2, 6, 0 ]
Hence G is infinite.
true
```
##### `LowIndexNormalSubgroups(G, n: parameters): GrpFP, RngIntElt -> [ Rec ]`
```magma
PrintLevel: RingIntElt Default: 0
Simplify : MonStgElt Default: "No"
Workspace : RngIntElt Default: 1000000000
```
The normal subgroups of finitely presented group $G$ up to index $n$, $n \le 100\,000$. The subgroups are returned as a sequence of records (ordered by subgroup index) where the $i$th record contains fields
`Group`: A presentation of the $i$th normal subgroup.
`Index`: The index of the $i$th normal subgroup in $G$.
`Supergroups`: The set of positions in the sequence of the groups which are supergroups of the $i$th group.
This parameter may be set to 0, 1, or 2. At 0, the function prints no diagnostic output. At level 1, it outputs details of each normal subgroup being tested for further normal subgroups. Level 2 gives details of each test being performed on each normal subgroup.
This sets the `Workspace` parameter for calls of `pQuotient`.
The possible values are `"No"`, `"Yes"`, and `"LengthLimit"`. This determines the parameter values passed to the `Rewrite(G,H)` function, when this function is used. The value `"No"` sets parameter `Simplify:=false`. The value `"Yes"` sets parameter `Simplify:=true`. The value `"LengthLimit"` sets parameter `LengthLimit:=Index(G,H)`.
#### Operations for Subgroups of Finite Index
Most operations described in this subsection require a closed coset table for at least one subgroup of an fp-group. If a closed coset table is needed and has not been computed, a coset enumeration will be invoked. If the coset enumeration does not produce a closed coset table, a runtime error is reported.
Experienced users can control the behaviour of such indirectly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `H ^ u: GrpFP, GrpFPElt -> GrpFP`
##### `Conjugate(H, u): GrpFP, GrpFPElt -> GrpFP`
Given an fp-group $H$ and a word $u$ in an fp-group $K$, such that $H$ and $K$ are subgroups of some common fp-group $G$ and words in terms of the generators of $G$ are known for the generators of both $H$ and $K$, construct the subgroup of $G$ obtained by conjugating $H$ by $u$.
##### `H meet K: GrpFP, GrpFP -> GrpFP`
Given subgroups $H$ and $K$, both of finite index in some fp-group $G$, return the subgroup which is the intersection of $H$ and $K$.
This function requires closed coset tables for both, $H$ and $K$ in $G$.
##### `Core(G, H): GrpFP, GrpFP -> GrpFP`
Given a subgroup $H$ of finite index in the fp-group $G$, construct the core of $H$ in $G$.
This function requires a closed coset table for $H$ in $G$.
##### `GeneratingWords(G, H): GrpFP, GrpFP -> { GrpFPElt }`
Given a subgroup $H$ of the fp-group $G$, this function returns a set of words in the generators of $G$, generating $H$ as a subgroup of $G$ (assuming such words are known or can be constructed). Note that the returned generating set does not necessarily correspond to the internal generators of $H$. In particular, generating words obtained using the function `GeneratingWords` cannot be used to coerce elements from $H$ to $G$.
##### `MaximalOvergroup(G, H): GrpFP, GrpFP -> GrpFP`
Given a subgroup $H$ of finite index in the fp-group $G$, construct a maximal overgroup of $H$ in $G$. A *maximal overgroup* of $H$ is a maximal subgroup of $G$ that contains $H$. If $H$ is already maximal, the group $G$ is returned.
This function requires a closed coset table for $H$ in $G$.
##### `MinimalOvergroup(G, H): GrpFP, GrpFP -> GrpFP`
Given a subgroup $H$ of finite index in the fp-group $G$, construct a minimal overgroup of $H$ in $G$. A *minimal overgroup* of a subgroup $H$ is a subgroup $K$ of $G$ such that $K$ contains $H$ as a maximal subgroup. If $H$ is already maximal in $G$, the group $G$ is returned.
This function requires a closed coset table for $H$ in $G$.
##### `H ^ G: GrpFP, GrpFP -> GrpFP`
##### `NormalClosure(G, H): GrpFP, GrpFP -> GrpFP`
Given a subgroup $H$ of finite index in the fp-group $G$, construct the normal closure of $H$ in $G$.
This function requires a closed coset table for $H$ in $G$.
##### `Normaliser(G, H): GrpFP, GrpFP -> GrpFP`
##### `Normalizer(G, H): GrpFP, GrpFP -> GrpFP`
Given a subgroup $H$ of finite index in the fp-group $G$, construct the normaliser of $H$ in $G$. For a sample application of this function, see Example [Implicit Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-de08eb).
This function requires a closed coset table for $H$ in $G$.
##### `SchreierGenerators(G, H : parameters): GrpFP, GrpFP -> { GrpFPElt }`
```magma
Simplify: BoolElt Default: true
```
Given a subgroup $H$ of finite index in the fp-group $G$, return the Schreier generators for $H$ as a set of words in $G$.
If the parameter `Simplify` is set to `true` (default), a heuristic method of eliminating redundant Schreier generators is applied. To switch this feature off, set `Simplify` to `false`.
This function requires a closed coset table for $H$ in $G$.
##### `SchreierSystem(G, H): GrpFP, GrpFP -> {@ GrpFPElt @}, Map`
##### `Transversal(G, H): GrpFP, GrpFP -> {@ GrpFPElt @}, Map`
Given a subgroup $H$ of finite index in the fp-group $G$, construct a (right) Schreier system of coset representatives for $H$ in $G$. The function returns
(a) the Schreier system as a set of words in $G$;
(b) the corresponding Schreier coset function.
This function requires a closed coset table for $H$ in $G$.
##### `Transversal(G, H, K): GrpFP, GrpFP, GrpFP -> {@ GrpFPElt @}, Map`
Given subgroups $H$ and $K$, both of finite index in the fp-group $G$, return an indexed set of words which comprise a set of representatives for the double cosets $HuK$ of $H$ and $K$ in $G$, as well as a map from $G$ to the representatives. It should be noted that this function is evaluated by first constructing the right cosets of $H$ in $G$ and then computing the orbits of the cosets under the action of the generators of the subgroup $K$.
This function requires a closed coset table for $H$ in $G$.
##### `Example: Subgroup Constructions (ex-793825)`
We illustrate some of the subgroup constructions by using them to construct subgroups of small index in the two-dimensional space group p4g which has the presentation
$$
< r, s\, |\, r^2, s^4, (r,s)^2 >.
$$
```magma
> p4g := FPGroup< r, s | r^2 = s^4 = (r*s^-1*r*s)^2 = 1 >;
> p4g;
Finitely presented group p4g on 2 generators
Relations
r^2 = Id(p4g)
s^4 = Id(p4g)
(r * s^-1 * r * s)^2 = Id(p4g)
```
We define two subgroups of $p4g$ and compute their indices in $p4g$.
```magma
> h := sub< p4g | (s^-1*r)^4, s*r >;
> k := sub< p4g | (s^-1*r)^2, (s*r)^2 >;
> Index(p4g, h);
8
> Index(p4g, k);
8
```
We construct the normal closure of $h$ in $p4g$.
```magma
> n := NormalClosure(p4g, h);
> n;
Finitely presented group n on 6 generators
Index in group p4g is 2
Generators as words in group p4g
n.1 = (s^-1 * r)^4
n.2 = s * r
n.3 = r * s
n.4 = r^-1 * s * r^2
n.5 = s^2 * r * s^-1
n.6 = r * s
```
Next, we construct a subgroup of $p4g$ containing $h$ as maximal subgroup…
```magma
> m := MinimalOvergroup(p4g, h);
> m;
Finitely presented group m on 3 generators
Index in group p4g is 4 = 2^2
Generators as words in group p4g
m.1 = (s^-1 * r)^4
m.2 = s * r
m.3 = (r * s)^2
```
…and a maximal subgroup of $p4g$ containing $k$.
```magma
> n := MaximalOvergroup(p4g, k);
> n;
Finitely presented group n on 4 generators
Index in group p4g is 2
Generators as words in group p4g
n.1 = (s^-1 * r)^2
n.2 = (s * r)^2
n.3 = r
n.4 = s^2
```
Finally, we construct a transversal in $p4g$ for the normaliser of $h$ in $p4g$…
```magma
> T := Transversal(p4g, Normaliser(p4g, h));
> T;
{@ Id(p4g), r, s^-1, r * s @}
```
…compute the intersection of $h$ and the conjugate of $h$ by $r$…
```magma
> l := h meet h^r;
> l;
Finitely presented group l
Index in group p4g is 32 = 2^5
Subgroup of group p4g defined by coset table
```
…and construct the core of $h$ in $p4g$.
```magma
> c := Core(p4g, h);
> c;
Finitely presented group c
Index in group p4g is 32 = 2^5
Subgroup of group p4g defined by coset table
```
Note, that the two subgroups $l$ and $c$ constructed last are defined as finite index subgroups of $p4g$ by a coset table and that there are no generators known for them. Generators can be obtained e.g. by using the function [`GeneratingWords`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-generatingwords). We show this for the subgroup $l$.
```magma
> GeneratingWords(p4g, l);
{ (s * r)^4, (s^-1 * r)^4 }
```
Once computed, these generators are memorised. Compare the result of printing $l$ to the output obtained above.
```magma
> l;
Finitely presented group l on 2 generators
Index in group p4g is 32 = 2^5
Generators as words in group p4g
l.1 = (s * r)^4
l.2 = (s^-1 * r)^4
```
##### `Example: Schreier Generators (ex-f2320c)`
Consider the group $G$ given by the presentation $\langle x,y\, |\, x^2, y^3, (xy)^7\rangle$.
```magma
> G := FPGroup< x,y | x^2, y^3, (x*y)^7 >;
```
We construct the subgroups of index less than or equal to 7 using the low index algorithm.
```magma
> L := LowIndexSubgroups(G, 7);
> L;
[
Finitely presented group on 2 generators
Index in group G is 1
Generators as words in group G
$.1 = x
$.2 = y,
Finitely presented group on 3 generators
Index in group G is 7
Generators as words in group G
$.1 = x
$.2 = y * x * y^-1
$.3 = y^-1 * x * y^-1 * x * y * x * y,
Finitely presented group on 3 generators
Index in group G is 7
Generators as words in group G
$.1 = x
$.2 = y * x * y^-1
$.3 = y^-1 * x * y * x * y^-1 * x * y
]
```
We define a subgroup as the core of one of the subgroups of index 7. The function [`Core`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-core) returns a subgroup of $G$ defined by a coset table.
```magma
> H := Core(G, L[2]);
> H;
Finitely presented group H
Index in group G is 168 = 2^3 * 3 * 7
Subgroup of group G defined by coset table
```
A set of generators for $H$ can be obtained e.g. with the function [`SchreierGenerators`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-schreiergenerators).
```magma
> sgH := SchreierGenerators(G, H);
> #sgH;
6
```
By default, [`SchreierGenerators`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-schreiergenerators) returns a reduced generating set. The unreduced set of Schreier generators can be obtained by setting the value of the parameter `Simplify` to `false`.
```magma
> sgHu := SchreierGenerators(G, H : Simplify := false);
> #sgHu;
85
```
#### Properties of Subgroups
The operations described in this subsection all require a closed coset table for at least one subgroup of an fp-group. If a closed coset table is needed and has not been computed, a coset enumeration will be invoked. If the coset enumeration does not produce a closed coset table, a runtime error is reported.
Experienced users can control the behaviour of such indirectly invoked coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
##### `u in H: GrpFPElt, GrpFP -> BoolElt`
Given an fp-group $H$ and a word $u$ in an fp-group $K$, such that $H$ and $K$ are subgroups of some common fp-group $G$, $H$ is of finite index in $G$, and words for the generators of $K$ in terms of the generators of $G$ are known, return `true` if $u$ is an element of $H$ and `false` otherwise.
This function requires a closed coset table for $H$ in $G$.
##### `u notin H: GrpFPElt, GrpFP -> BoolElt`
Given an fp-group $H$ and a word $u$ in an fp-group $K$, such that $H$ and $K$ are subgroups of some common fp-group $G$, $H$ is of finite index in $G$, and words for the generators of $K$ in terms of the generators of $G$ are known, return `true` if $u$ is not an element of $H$ and `false` otherwise.
This function requires a closed coset table for $H$ in $G$.
##### `H eq K: GrpFP, GrpFP -> BoolElt`
Given subgroups $H$ and $K$, both of finite index in the fp-group $G$, return `true` if $H$ and $K$ are equal and `false` otherwise.
This function may require closed coset tables for both, $H$ and $K$ in $G$.
##### `H ne K: GrpFP, GrpFP -> BoolElt`
Given subgroups $H$ and $K$, both of finite index in the fp-group $G$, return `true` if $H$ and $K$ are not equal and `false` otherwise.
This function may require closed coset tables for both, $H$ and $K$ in $G$.
##### `H subset K: GrpFP, GrpFP -> BoolElt`
##### `IsSubgroup(H, K): GrpFP, GrpFP -> BoolElt`
Given subgroups $H$ and $K$, both of finite index in the fp-group $G$, return `true` if $H$ is contained in $K$ and `false` otherwise.
This function requires a closed coset table for $K$ in $G$.
##### `H notsubset K: GrpFP, GrpFP -> BoolElt`
Given subgroups $H$ and $K$, both of finite index in the fp-group $G$, return `true` if $H$ is not contained in $K$ and `false` otherwise.
This function requires a closed coset table for $K$ in $G$.
##### `IsConjugate(G, H, K): GrpFP, GrpFP, GrpFP -> BoolElt, GrpFPElt`
Given subgroups $H$ and $K$, both of finite index in the fp-group $G$, return `true` if $H$ and $K$ are conjugate subgroups of $G$ and `false` otherwise. If $H$ and $K$ are conjugate in $G$, a conjugating element is returned as second return value.
This function requires a closed coset table for both, $H$ and $K$ in $G$.
##### `IsNormal(G, H): GrpFP, GrpFP -> BoolElt`
Given a subgroup $H$ of finite index in the fp-group $G$, return `true` if $H$ is a normal subgroup of $G$ and `false` otherwise.
This function requires a closed coset table for $H$ in $G$.
##### `IsMaximal(G, H): GrpFP, GrpFP -> BoolElt`
Given a subgroup $H$ of finite index in the fp-group $G$, return `true` if $H$ is a maximal subgroup of $G$ and `false` otherwise.
This function requires a closed coset table for $H$ in $G$.
##### `IsSelfNormalizing(G, H): GrpFP, GrpFP -> BoolElt`
Given a subgroup $H$ of finite index in the fp-group $G$, return `true` if $H$ is a self-normalizing subgroup of $G$ and `false` otherwise. For a sample application of this function, see Example [Implicit Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#example-ex-de08eb).
This function requires a closed coset table for $H$ in $G$.
##### `Example: Subgroup Predicates in a Space Group (ex-afcd46)`
We illustrate some of the subgroup predicates by applying them to some subgroups of the two-dimensional space group $p4g = < r, s | r^2, s^4, (r,s)^2 >$ from Example [Subgroup Constructions](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#example-ex-793825).
```magma
> p4g := FPGroup< r, s | r^2 = s^4 = (r*s^-1*r*s)^2 = 1 >;
> h := sub< p4g | (s^-1*r)^4, s*r >;
> k := sub< p4g | (s^-1*r)^2, (s*r)^2 >;
```
$h$ and $k$ have the same index in $p4g$…
```magma
> Index(p4g, h);
8
> Index(p4g, k);
8
```
…but they are not equal.
```magma
> h eq k;
false
```
We check for normality of $h$ and $k$ in $p4g$.
```magma
> IsNormal(p4g, h);
false
> IsNormal(p4g, k);
true
```
We construct the normal closure of $h$ in $p4g$.
```magma
> n := NormalClosure(p4g, h);
```
We see that it is maximal…
```magma
> IsMaximal(p4g, n);
true
```
…and that it contains $k$.
```magma
> k subset n;
true
```
We define another subgroup of $p4g$.
```magma
> l := sub< p4g | (s*r)^4, s^-1*r >;
```
In fact, it is conjugate to $h$.
```magma
> IsConjugate(p4g, h, l);
true r^-1
```
i.e. $l = h^{r^{-1}}$. The intersection of $h$ and $h^{r^{-1}}$ already yields the core of $h$ in $p4g$.
```magma
> h meet l eq Core(p4g, h);
true
```
##### `Example: Maximal Subgroups of the Hall-Janko Group (ex-bf7b3f)`
The constructions of the previous section together with the Boolean function [`IsMaximal`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-ismaximal) may be used to locate large maximal subgroups in a finite group. Consider the Hall-Janko group $J_2$, which may be defined by the presentation
$$
\begin{aligned}.\end{aligned}
$$
We examine subgroups generated by pairs of randomly chosen short words. Whenever we obtain a proper subgroup, if it is not already maximal we replace it by a maximal subgroup that contains it.
```magma
> J2 := FPGroup (c*b)^5, (c*b^-1*c*b)^2,
> a^-1*b*a*c*a^-1*b*a*c^-1*a^-1*b^-1*a*c^-1,
> a*b*a^-1*c*a*b*a^-1*c^-1*a*b^-1*a^-1*c^-1>;
>
> Seen := { 0, 1};
> Found := { };
> Sgs := [ ];
> for i := 1 to 30 do
> u := Random(J2, 1, 1);
> v := Random(J2, 3, 5);
> H := sub< J2 | u, v >;
> Indx := Index(J2, H);
> if Indx notin Seen then
> Include(~Seen, Indx);
> if not IsMaximal(J2, H) then
> H := MaximalOvergroup(J2, H);
> end if;
> if Indx notin Found then
> Include(~Sgs, H);
> Include(~Seen, Indx);
> Include(~Found, Indx);
> end if;
> end if;
> end for;
> Sgs;
[
Finitely presented group on 3 generators
Index in group J2 is 315 = 3^2 * 5 * 7
Generators as words in group J2
$.1 = b^-1
$.2 = a^-2 * c * a^-1
$.3 = c,
Finitely presented group on 3 generators
Index in group J2 is 1008 = 2^4 * 3^2 * 7
Generators as words in group J2
$.1 = b^-1
$.2 = c^-1 * a^-1 * c^-1 * b
$.3 = a * c * b * c^-1 * a^-1 * c * b * c^-1,
Finitely presented group on 3 generators
Index in group J2 is 100 = 2^2 * 5^2
Generators as words in group J2
$.1 = c
$.2 = (b * a^-1)^2
$.3 = a * b^-1
]
```
Thus after taking $30$ $2$-generator random subgroups, we have obtained three maximal subgroups, including the two largest maximal subgroups.
### Finite FP-Groups
In this section a number of intrinsics that apply only to finite fp-group will be described. Of course most of the intrinsics designed for working with general fp-groups will also apply to finite fp-groups.
#### Concrete Representations
If a finitely-presented group is known to have moderate order, it may be useful to write it concretely as a permutation or PC-group. There are two utilities provided for this purpose.
##### `PermutationGroup(G): GrpFP -> GrpPerm, GrpHom`
Construct a faithful permutation representation of $G$, and the isomorphism from $G$ to the representation. The algorithm first computes the order of $G$, then takes the regular representation of $G$, and then tries to reduce the degree. Thus, the intrinsic is restricted to groups of order a million (depending upon the amount of memory available).
##### `PCGroup(G): GrpFP -> GrpPC, GrpHom`
Construct a faithful PC-group representation of $G$, and the isomorphism from $G$ to the representation. The algorithm first computes the order of $G$, then computes the soluble quotient of $G$ with the order found. Thus the command is restricted to small soluble groups.
#### The Cayley Graph
Given a finite fp-group $G$ of moderate order we can construct its Cayley graph. This graph can then be studied using the extensive tools provided in Magma for graphs.
##### `CayleyGraph(G): GrpFP -> GrphVertSet, GrphEdgeSeto`
```magma
Labelled: BoolElt Default Default: true
```
Given a finite group $G$ defined on generating set $X$, construct the Cayley graph $C$ of $G$ relative to the generating set $X$. This graph is defined as follows: The vertices correspond to the elements of $G$ and two vertices $u$, $v$ are adjacent if and only if there exists an generator $x$ in $X$ such that $u * x = v$.
The optional parameter Labelled (Labelled := true by default) can be set to false to prevent the graph being labelled. If this is not done, then the vertices of $C$ will be labelled with the appropriate elements of $G$ and the (directed) edge from $u$ to $v$ will be labelled with the appropriate generator $x$ as defined above.
##### `SchreierGraph(G, H): Grp, Grp -> Grph, GrphVertSet, GrphEdgeSet`
```magma
Labelled: BoolElt Default Default: true
Directed: BoolElt Default Default: true
```
Given a finite fp-group $G$ defined on the generating set $X$ and a subgroup $H$ of $G$, construct the Schreier coset graph $S$ for $G$ over $H$, relative to $X$. The graph $S$ is defined as follows: The vertices correspond to the cosets of $H$ in $G$, and two vertices $u$, $v$ are adjacent in $S$ if and only if there exists an element $x$ in $X$ such that $u * x = v$.
The graph returned may be a labelled or unlabelled version and directed or undirected. These versions are controlled by the parameters `Labelled` and `Directed`, which are both true by default.
##### `Example: Cayley Graph of a Dihedral Group (ex-6ec7fa)`
We define the Cayley graph of the dihedral group of order 12 and then construct its directed Cayley graph.
```magma
> G := DihedralFPGroup(6);
> C, V, E := CayleyGraph(G); C;
Digraph
Vertex Neighbours
1 2 3 ;
2 5 6 ;
3 1 7 ;
4 1 7 ;
5 9 10 ;
6 2 3 ;
7 4 11 ;
8 4 11 ;
9 8 12 ;
10 5 6 ;
11 8 12 ;
12 9 10 ;
```
We next find the diameter $d$ of the graph and then construct the word corresponding to a path of length $d$.
```magma
> d := Diameter(C); d;
4
> p := DiameterPath(C); p;
[ 1, 2, 5, 9, 8 ]
> word := &* [ Label(E![p[i], p[i+1]]) : i in [1..d]];
> word;
x^4
```
We next construct the automorphism group of the unlabelled, undirected graph underlying $C$.
```magma
> D := UnderlyingGraph(C); D;
Graph
Vertex Neighbours
1 2 3 4 ;
2 1 5 6 ;
3 1 6 7 ;
4 1 7 8 ;
5 2 9 10 ;
6 2 3 10 ;
7 3 4 11 ;
8 4 9 11 ;
9 5 8 12 ;
10 5 6 12 ;
11 7 8 12 ;
12 9 10 11 ;
> A := AutomorphismGroup(D); A;
Permutation group A acting on a set of cardinality 12
Order = 24 = 2^3 * 3
(2, 4)(5, 8)(6, 7)(10, 11)
(1, 2)(3, 6)(4, 5)(7, 10)(8, 9)(11, 12)
(1, 3)(2, 6)(4, 7)(5, 10)(8, 11)(9, 12)
```
Finally we check that $S$ has the dihedral group of order 12 as a subgroup.
```magma
> subs := Subgroups(A);
> P := PermutationGroup(G);
> exists(j){ i : i in [1..#subs] | IsIsomorphic(P, subs[i]`subgroup)};
true
> j;
25
```
So subgroup 25 is isomorphic to G. In fact there are four conjugacy classes of subgroups in $A$ that are isomorphic to $G$.
### Homomorphisms
For a general description of homomorphisms, we refer to Chapter [Mappings](https://docs.magma-maths.org/SetsSequencesMappings/Mappings/index-mappings.md#chapmap). This section describes some special aspects of homomorphisms the domain of which is a finitely presented group.
#### General Remarks
The kernel of a homomorphism with a domain of type `GrpFP` can be computed using the function [`Kernel`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-kernel), if the codomain is of one of the types `GrpGPC`, `GrpPC` (cf. Chapter [Finite Soluble Groups](https://docs.magma-maths.org/FiniteGroups/FiniteSolubleGroups/index-finite-soluble-groups.md#chapgrppc)), `GrpAb` (cf. Chapter [Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)), `GrpPerm` (cf. Chapter [Permutation Groups](https://docs.magma-maths.org/FiniteGroups/PermutationGroups/index-permutation-groups.md#chapgrpperm)), `GrpMat` (cf. Chapter [Matrix Groups over General Rings](https://docs.magma-maths.org/FiniteGroups/MatrixGroupsOverGeneralRings/index-matrix-groups-over-general-rings.md#chapgrpmat)), `ModAlg`, or `ModGrp` (cf. Chapter [Modules over an Algebra and Group Representations](https://docs.magma-maths.org/RepresentationTheory/ModulesOverAnAlgebraAndGroupRepresentations/index-modules-over-an-algebra-and-group-representations.md#chapmodalg)), if the image is finite and its order sufficiently small. In this case, a regular permutation representation of the image is constructed and the kernel is created as a subgroup of the domain, defined by a coset table.
The kernel may also be computable, if the codomain is of the type `GrpFP`, the image is sufficiently small, and a presentation for the image is known.
If the kernel of a map can be computed successfully, forming preimages of substructures is possible. An attempt to compute the kernel of a map will be made automatically, if the preimage of a substructure of the codomain is to be computed.
Note, that trying to compute the kernel may be very time and memory consuming; use this feature with care.
#### Construction of Homomorphisms
##### `hom< P -> G | S >: Struct, Struct -> Map`
Returns the homomorphism from the fp-group $P$ to the group $G$ defined by the assignment $S$. $S$ can be the one of the following:
**(i)**
A list, sequence, or indexed set containing the images of the $n$ generators $P.1,\ldots,P.n$ of $P$. Here, the $i$-th element of $S$ is interpreted as the image of $P.i$, i.e. the order of the elements in $S$ is important.
**(ii)**
A list, sequence, enumerated set, or indexed set, containing $n$ tuples $$ or arrow pairs $x_i \;-\!\!> y_i$, where $x_i$ is a generator of $P$ and $y_i\in G$ ($i=1,\ldots,n$) and the set $\{x_1,\ldots,x_n\}$ is the full set of generators of $P$. In this case, $y_i$ is assigned as the image of $x_i$, hence the order of the elements in $S$ is not important.
Note, that it is currently not possible to define a homomorphism by assigning images to the elements of an arbitrary generating set of $P$. It is the user’s responsibility to ensure that the arguments passed to the `hom`-constructor actually yield a well-defined homomorphism. For certain codomain categories, this may be checked using the function [`IsSatisfied`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-issatisfied) described below.
##### `IsSatisfied(U, E): { RelElt }, [ GrpElt ] -> BoolElt`
##### `IsSatisfied(U, E): { GrpFPElt }, [ GrpElt ] -> BoolElt`
##### `IsSatisfied(U, E): [ RelElt ], [ GrpElt ] -> BoolElt`
##### `IsSatisfied(U, E): [ GrpFPElt ], [ GrpElt ] -> BoolElt`
The argument $U$ is a set or sequence of either words belonging to an $n$-generator fp-group $H$ or relations over $H$. The second argument $E$ is a sequence of $n$ elements $[e_1, \ldots, e_n]$ belonging to a group $G$ for which both, multiplication and comparison of elements are possible. Using the mapping $H.i \rightarrow e_i$ ($i = 1, \ldots, n$), we evaluate the relations given by $U$. If $U$ is a set or sequence of relations, the left and right hand sides of each relation are evaluated and compared for equality. Otherwise, each word in $U$ is evaluated and compared to the identity. If all relations are satisfied, `IsSatisfied` returns the Boolean value `true`. On the other hand, if any relation is not satisfied, `IsSatisfied` returns the value `false`.
This function may be used to verify the correctness of the definition of a homomorphism from an fp-group to a group in a category for which both, multiplication and comparison of elements are possible.
#### Accessing Homomorphisms
##### `w @ f: GrpFPElt, Map -> GrpElt`
##### `f(w): Map, GrpFPElt -> GrpElt`
Given a homomorphism whose domain is an fp-group $G$ and an element $w$ of $G$, return the image of $w$ under $f$ as an element of the codomain of $f$.
##### `H @ f: GrpFP, Map -> Grp`
##### `f(H): Map, GrpFP -> Grp`
Given a homomorphism whose domain is an fp-group $G$ and a subgroup $H$ of $G$, return the image of $H$ under $f$ as a subgroup of the codomain of $f$.
Some maps do not support images of subgroups.
##### `g @@ f: GrpElt, Map -> GrpFPElt`
Given a homomorphism whose domain is an fp-group $G$ and an element $g$ of the image of $f$, return the preimage of $g$ under $f$ as an element of $G$.
Some maps do not support inverse images.
##### `H @@ f: Grp, Map -> GrpFP`
Given a homomorphism whose domain is an fp-group $G$ and a subgroup $H$ of the image of $f$, return the preimage of $H$ under $f$ as a subgroup of $G$.
Some maps do not support inverse images. The inverse image of a subgroup of the codomain can only be computed if the kernel of the homomorphism can be computed, i.e. if the kernel has moderate index in the domain.
##### `Domain(f): Map -> Grp`
The domain of the homomorphism $f$.
##### `Codomain(f): Map -> Grp`
The codomain of the homomorphism $f$.
##### `Image(f): Map -> Grp`
The image or range of the homomorphism $f$ as a subgroup of the codomain of $f$.
Some maps do not support this function.
##### `Kernel(f): Map -> Grp`
The kernel of the homomorphism $f$ as a (normal) subgroup of the domain of $f$, represented by a coset table.
Some maps do not support this function. The kernel of a homomorphism can only be computed, if it has moderate index in the domain.
##### `Example: Braid Group onto Sym(5) (ex-63663e)`
For arbitrary $n>0$, the symmetric group of degree $n+1$ is an epimorphic image of the braid group on $n$ generators. In this example, we exhibit this relationship for $n=4$.
We start with creating the braid group $B$ on 5 strings, i.e. 4 Artin generators.
```magma
> B := BraidFPGroup(5);
> B;
Finitely presented group B on 4 generators
Relations
B.1 * B.2 * B.1 = B.2 * B.1 * B.2
B.1 * B.3 = B.3 * B.1
B.1 * B.4 = B.4 * B.1
B.2 * B.3 * B.2 = B.3 * B.2 * B.3
B.2 * B.4 = B.4 * B.2
B.3 * B.4 * B.3 = B.4 * B.3 * B.4
```
In the symmetric group of degree 5, we define 4 transpositions which will be the images of the generators of $B$.
```magma
> S := Sym(5);
> imgs := [ S!(1,2), S!(2,3), S!(3,4), S!(4,5) ];
```
In order to verify that this assignment actually gives rise to a well defined homomorphism, we check whether the potential images satisfy the defining relations of $B$.
```magma
> rels := Relations(B);
> rels;
[ B.1 * B.2 * B.1 = B.2 * B.1 * B.2, B.1 * B.3 = B.3 * B.1,
B.1 * B.4 = B.4 * B.1, B.2 * B.3 * B.2 = B.3 * B.2 * B.3,
B.2 * B.4 = B.4 * B.2, B.3 * B.4 * B.3 = B.4 * B.3 * B.4 ]
> IsSatisfied(rels, imgs);
true
```
They do. So we can define the homomorphism from $B$ to $S$.
```magma
> f := hom< B->S | imgs >;
```
We see that $f$ is surjective, i.e. $S$ is an epimorphic image of $B$ as claimed above.
```magma
> f(B) eq S;
true
```
We now check the kernel of $f$.
```magma
> Kernel(f);
Finitely presented group
Index in group B is 120 = 2^3 * 3 * 5
Subgroup of group B defined by coset table
```
Using the function [`GeneratingWords`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-fin-index.md#function-grpfp-1-generatingwords) described later, we can obtain a set of generators of $\ker(f)$ as a subgroup of $B$.
```magma
> GeneratingWords(B, Kernel(f));
{ B.2^-2, (B.1 * B.2 * B.3^-1 * B.2^-1 * B.1^-1)^2, B.1^-2,
(B.3 * B.4^-1 * B.3^-1)^2, (B.2 * B.3^-1 * B.2^-1)^2,
(B.2 * B.3 * B.4^-1 * B.3^-1 * B.2^-1)^2, B.4^-2,
(B.1 * B.2^-1 * B.1^-1)^2, B.3^-2,
(B.1 * B.2 * B.3 * B.4^-1 * B.3^-1 * B.2^-1 * B.1^-1)^2 }
```
It is easy to see that all generators of $\ker(f)$ are conjugates of words of the form $g^{\pm2}$, where $g$ is a generator of $B$. We check this, using the normal closure constructor [`ncl< G | L >`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#constructor-grpfp-1-ncl) described later.
```magma
> Kernel(f) eq ncl< B | B.1^2, B.2^2, B.3^2, B.4^2 >;
true
```
Thus, the braid relations together with the relations $B.1^2, B.2^2, B.3^2, B.4^2$ are a set of defining relations for $S$.
#### Constructing Homomorphisms onto Finite Groups
This section describes functions for computing representatives of the classes of homomorphisms from a finitely presented group $F$ to a finite group $G$ modulo a group $A$ of automorphisms of $G$.
##### `Homomorphisms(F, G, A : parameters): GrpFP, GrpPerm, GrpPerm -> [ HomGrp ]`
##### `Homomorphisms(F, G : parameters): GrpFP, GrpPerm -> [ HomGrp ]`
```magma
Surjective : BoolElt Default: true
Limit : RngIntElt Default: 0 (no limit)
TimeLimit : RngIntElt Default: 0 (no limit)
CosetEnumeration: BoolElt Default: true
CacheCosetAction: BoolElt Default: true
```
Given a finitely presented group $F$ and two permutation groups $G$ and $A$ with $G\triangleleft A$, return a sequence containing representatives of the classes of homomorphisms from $F$ to $G$ modulo automorphisms of $G$ induced by elements of $A$. (That is, two homomorphisms $f_1$, $f_2 : F \rightarrow G$ are considered equivalent if there exists an element $a\in A$ such that $f_1(x) = f_2(x)^a$ for all $x\in F$.) The call `Homomorphisms(F, G)` is equivalent to `Homomorphisms(F, G, G)`. The function uses a backtrack algorithm testing certain conjugates of representatives of the $A$-classes in $G$ as possible images of the generators of $F$. The following parameters are available for this function.
If this parameter is set to `true` (default), only epimorphisms are considered.
If this parameter is set to $n$, the function terminates after $n$ classes of homomorphisms satisfying the specified conditions have been found. A value of 0 (default) means no limit.
A limit in seconds for the amount of time spent in the backtrack search for homomorphisms. The time spent in initial coset enumerations is *not* counted towards this limit. A value of 0 (default) means no limit.
If this parameter is set to `true` (default), a number of short coset enumerations are performed for each generator of $F$ in order to check whether some $A$-classes in $G$ can be ruled out as possible images for this generator. If a class can be ruled out, the complexity of the backtrack search may be reduced significantly. In situations where it seems unlikely that classes can be ruled out as possible generator images, experienced users may wish to turn this feature off in order to save the time spent on the coset enumerations.
The value of this parameter indicates whether the actions of $A$ on the cosets of the centralisers of representatives of the $A$-classes in $G$ are cached during the backtrack search. Setting this parameter to `true` (default) results in faster computations and is recommended for normal applications. When computing homomorphisms to large groups with many conjugate classes, this parameter can be set to `false` in order to reduce memory requirements at the expense of increased computing time.
##### `Example: Finding Homomorphisms onto Finite Groups (ex-731266)`
Consider the finitely presented group
$$
F\ := \ < a,b,c \ |\ ac^{-1}bc^{-1}aba^{-1}b,
abab^{-1}c^2b^{-1},
a^2b^{-1}(ca)^4cb^{-1}>.
$$
```magma
> F := FPGroup< a,b,c | a*c^-1*b*c^-1*a*b*a^-1*b,
> a*b*a*b^-1*c^2*b^-1,
> a^2*b^-1*c*a*c*a*c*a*c*a*c*b^-1 >;
```
We use the function [`Homomorphisms`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-homomorphisms) to prove that $F$ maps onto $A_5$.
```magma
> G := Alt(5);
> homs := Homomorphisms(F, G : Limit := 1);
> #homs gt 0;
true
```
##### `Homomorphisms(F, G, A : parameters): GrpFP, GrpPC, GrpPC -> [ HomGrp ]`
##### `Homomorphisms(F, G : parameters): GrpFP, GrpPC -> [ HomGrp ]`
```magma
Surjective: BoolElt Default: true
Limit : RngIntElt Default: 0 (no limit)
```
Given a finitely presented group $F$ and two finite polycyclic groups $G$ and $A$ with $G\triangleleft A$, return a sequence containing representatives of the classes of homomorphisms from $F$ to $G$ modulo automorphisms of $G$ induced by elements of $A$. (That is, two homomorphisms $f_1$, $f_2 : F \rightarrow G$ are considered equivalent if there exists an element $a\in A$ such that $f_1(x) = f_2(x)^a$ for all $x\in F$.) The call `Homomorphisms(F, G)` is equivalent to `Homomorphisms(F, G, G)`. The following parameters are available for this function.
If this parameter is set to `true` (default), only epimorphisms are considered.
If this parameter is set to $n$, the function terminates after $n$ classes of homomorphisms satisfying the specified conditions have been found. A value of 0 (default) means no limit.
The next group of intrinsics describe a process version of the algorithm described above which computes the homomorphisms one at a time. When searching for a quotient of the group having certain properties this technique can save a great deal of time.
##### `HomomorphismsProcess(F, G, A : parameters): GrpFP, GrpPerm, GrpPerm -> GrpFPHomsProc`
##### `HomomorphismsProcess(F, G : parameters): GrpFP, GrpPerm -> GrpFPHomsProc`
```magma
Surjective : BoolElt Default: true
Limit : RngIntElt Default: 0 (no limit)
TimeLimit : RngIntElt Default: 0 (no limit)
CosetEnumeration: BoolElt Default: true
CacheCosetAction: BoolElt Default: true
```
Given a finitely presented group $F$ and two permutation groups $G$ and $A$ with $G\triangleleft A$, return a process $P$ for computing representatives of the classes of homomorphisms from $F$ to $G$ modulo automorphisms of $G$ induced by elements of $A$. (That is, two homomorphisms $f_1$, $f_2 : F \rightarrow G$ are considered equivalent if there exists an element $a\in A$ such that $f_1(x) = f_2(x)^a$ for all $x\in F$.) `HomomorphismsProcess(F, G)` is equivalent to `HomomorphismsProcess(F, G, G)`. After constructing the process, the search for homomorphisms is started and runs until the first homomorphism is found, the time limit is reached (in which case $P$ is marked as *invalid*) or the search is completed without finding a homomorphism (in which case $P$ is marked as *empty*). The parameters have the same meaning as for the function [`Homomorphisms`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-homomorphisms).
Setting a time limit for a process $P$ limits the total amount of time spent in the backtrack search for homomorphisms during the construction of $P$ and in subsequent calls to [`NextElement`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#procedure-grpfp-1-nextelement). The time spent in initial coset enumerations is *not* counted towards this limit. If the time limit or the limit on the number of homomorphisms set for a process $P$ is reached, $P$ becomes *invalid*. Calling [`NextElement`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#procedure-grpfp-1-nextelement) for an invalid process or for a process which is *empty*, that is, which has found all possible classes of homomorphisms, will cause a runtime error. The functions [`IsValid`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isvalid-hom) and [`IsEmpty`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isempty-hom) can be used to check whether a process is valid or empty, respectively. The use of these functions is recommended to avoid runtime errors in loops or user written functions.
##### `NextElement(~P): GrpFPHomsProc`
Given a valid and non-empty process $P$, continue the backtrack search until a new class of homomorphisms is found. If the search completes without a new representative being found, $P$ is marked as empty. If a limit set for $P$ is reached, $P$ is marked as invalid.
##### `IsEmpty(P): GrpFPHomsProc -> BoolElt`
Returns whether $P$ is empty, that is, whether all possible classes of homomorphisms have been found.
##### `IsValid(P): GrpFPHomsProc -> BoolElt`
Returns `false` if a limit set for $P$ has been reached and `true` otherwise. Note that the return value of `IsValid` is not related to whether or not $P$ currently defines a homomorphism.
##### `DefinesHomomorphism(P): GrpFPHomsProc -> BoolElt`
Returns whether $P$ currently defines a homomorphism which can be extracted using the function [`Homomorphism`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-homomorphism).
##### `Homomorphism(P): GrpFPHomsProc -> HomGrp`
Given a process $P$ which defines a homomorphism, return this homomorphism, that is, the homomorphism most recently found by $P$. If $P$ does not define a homomorphism, a runtime error will result. The function [`DefinesHomomorphism`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-defineshomomorphism) can be used to test whether a call to `Homomorphism` is legal for a process.
##### `# P: GrpFPHomsProc -> RngIntElt`
Return the number of homomorphisms that have been found by the process $P$.
##### `Example: Braid Group onto PSL(2, 16) (ex-9b82f6)`
Consider the braid group $B$ on 4 strings.
We show how the interactive computation of homomorphisms can be used to determine a homomorphism $f: B \ar \PSL(2,16)$ whose image is a maximal subgroup of $\PSL(2,16)$. Note how the functions [`IsValid`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isvalid-hom), [`IsEmpty`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isempty-hom), and [`DefinesHomomorphism`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-defineshomomorphism) are used to avoid runtime errors in the `while` loop.
```magma
> B := BraidFPGroup(4);
> G := PSL(2,16);
> P := HomomorphismsProcess(B, G : Surjective := false,
> TimeLimit := 10);
> while not IsEmpty(P) do
> if DefinesHomomorphism(P) then
> f := Homomorphism(P);
> img := Image(f);
> if IsMaximal(G,img) then
> print "found image which is maximal subgroup";
> break;
> end if;
> end if;
> if IsValid(P) then
> NextElement(~P);
> else
> print "Limit has been reached";
> break;
> end if;
> end while;
found image which is maximal subgroup
>
> f;
Homomorphism of GrpFP: B into GrpPerm: G, induced by
B.1 |--> (1, 4, 3, 6, 12)(2, 11, 9, 16, 7)(5, 17, 8, 15, 13)
B.2 |--> (1, 4, 2, 10, 16)(3, 11, 9, 12, 14)(5, 15, 8, 13, 17)
B.3 |--> (1, 4, 3, 6, 12)(2, 11, 9, 16, 7)(5, 17, 8, 15, 13)
```
##### `Example: Proving Infiniteness Interactively (ex-3b7c60)`
This example sketches how the interactive version of the homomorphism algorithm could be used as part of a function trying to prove that a group is infinite. Note again how the functions [`IsValid`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isvalid-hom), [`IsEmpty`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-isempty-hom), and [`DefinesHomomorphism`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-defineshomomorphism) are used to avoid runtime errors.
```magma
> function MyIsInfinite(F)
>
> // ...
>
> // quotient approach: check whether an obviously infinite
> // normal subgroup can be found in reasonable time.
> S := [ Alt(5), PSL(2,7), PSL(2,9), PSL(2,11) ];
> for G in S do
> P := HomomorphismsProcess(F, G : Surjective := false,
> TimeLimit := 5);
> while IsValid(P) and not IsEmpty(P) do
> if DefinesHomomorphism(P) then
> f := Homomorphism(P);
> if 0 in AQInvariants(Kernel(f)) then
> print "found infinite normal subgroup";
> print "Hence group is infinite";
> return true;
> end if;
> end if;
> if IsValid(P) then
> NextElement(~P);
> end if;
> end while;
> end for;
> print "quotient approach failed; trying other strategies";
>
> // ...
>
> end function;
```
We try the code fragment on the group
$$
< a, b\ |\ ab^{-1}a^{-1}ba^{-1}b^{-1}abb,
ab^{-1}a^{-1}baaba^{-1}b^{-1}ab^{-1}a^{-1}baba^{-1}b^{-1} >.
$$
```magma
> F := FPGroup< a,b |
> a*b^-1*a^-1*b*a^-1*b^-1*a*b*b,
> a*b^-1*a^-1*b*a*a*b*a^-1*b^-1*a*b^-1*a^-1*b*a*b*a^-1*b^-1 >;
> MyIsInfinite(F);
found infinite normal subgroup
Hence group is infinite
true
```
#### Searching for Isomorphisms
This section describes a function for searching for isomorphisms between two finitely presented groups.
##### `SearchForIsomorphism(F, G, m : parameters): GrpFP, GrpFP, RngIntElt -> BoolElt, HomGrp, HomGrp`
```magma
All : BoolElt Default: false
IsomsOnly : BoolElt Default: true
MaxRels : RngIntElt Default: 250*m
CycConjTest: BoolElt Default: true
```
Attempt to find an isomorphism from the finitely presented group $F$ to the finitely presented group $G$. The search will be restricted to those homomorphisms for which the sum of the word-lengths of the images of the generators of $F$ in $G$ is at most $m$.
If an isomorphism $\phi$ is found, then the values `true`, $\phi$, $\phi^{-1}$ are returned. Otherwise, the values `false`, \_, \_ are returned; of course, that does not necessarily mean that the groups are not isomorphic.
An error will result if any of the generators of $F$ turn out to be trivial.
By setting the verbose flag `"IsoSearch"` to $1$, information about the progress of the search will be printed.
The parameters available for the function [`SearchForIsomorphism`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/homomorphism.md#function-grpfp-1-searchforisomorphism) are:
If `All` is `false` (default), then the function halts and returns as soon as a single isomorphism is found. If `All` is `true`, then the search continues through all possible images that satisfy the image length condition, and a list of pairs $\langle\phi,\phi^{-1}\rangle$ for all isomorphisms $\phi:F\to G$ found is returned as the second return value.
If `IsomsOnly` is set to `false`, then all homomorphisms $F \to G$ will be returned if `All` is `true`, and the first nontrivial homomorphism found will be returned if `All` is `false`.
The value of the `MaxRelations` parameter used in runs of [`RWSGroup`](https://docs.magma-maths.org/FinitelyPresentedGroups/GroupsDefinedByRewriteSystems/creation.md#function-grprws-rwsgroup). It is hard to find a sensible default, because if the value is unnecessarily large then time can be wasted unnecessarily. This may need to be increased if the function is used with finite groups, for example (although it is usually much more efficient to use permutation or matrix representations when testing isomorphism of finite groups).
When `CycConjTest` is `true` and `All` is `false`, then images of the first generator which have a cyclic conjugate that comes earlier in the lexicographical order are rejected, because there would be a conjugate isomorphism in which the image was the cyclic conjugate. This nearly always results in faster run-times, but occasionally it can happen that the conjugate isomorphism has a larger sum of lengths of generator images, which is clearly bad. So the user has the option of not rejecting such images.
##### `Example: Isomorphism Search for Hillman's Groups (ex-d9442c)`
John Hillman asked whether the following two groups are isomorphic.
```magma
> G1 := FPGroup u*(s*t)^2=(s*t)^2*u >;
> G2 := FPGroup y^2=(z^-1*x)^2, z*(x*y)^2=(x*y)^2*z >;
> isiso, f1, f2 := SearchForIsomorphism(G1,G2,7);
> isiso;
true
> f1;
Homomorphism of GrpFP: G1 into GrpFP: G2 induced by
s |--> x * z^-1
t |--> y * z
u |--> x * y^-1 * z
> f2;
Homomorphism of GrpFP: G2 into GrpFP: G1 induced by
x |--> s^2 * u * t^-1
y |--> s^-1 * u^-1
z |--> s * u * t^-1
```
The search for an isomorphism succeeded and returned the isomorphisms explicitly.
##### `Example: Isomorphism Search for Neumann's Groups (ex-4192a7)`
Walter Neumann asked whether the next two groups are isomorphic.
```magma
> G1 := FPGroup< x,y,z | x^2*y^5, x^14*z^23, (x^2,y), (x^2,z), x*y*z>;
> G2 := FPGroup;
```
It is a good idea to minimize the number of generators of `G1` - since there is clearly a redundant generator here.
```magma
> G1s := Simplify(G1);
> Ngens(G1s);
2
> G1!G1s.1, G1!G1s.2;
x z
```
Now a direct call `SearchForIsomorphism(G1, G2, 15)` will succeed, but will take many hours, and also use a lot of memory.
In contrast to `G1`, it can sometimes help to introduce a new generator in `G2` if there are common substrings in relators.
```magma
> G2b := FPGroup< a,b,c | a*b^16*a*b^-7, a^4*b^7*a^-1*b^7, c=b^7>;
> isiso, f1, f2 := SearchForIsomorphism(G1s,G2b,4);
> isiso;
true
> f1;
Homomorphism of GrpFP: G1s into GrpFP: G2b induced by
G1s.1 |--> a * c^-1
G1s.2 |--> c
> f2;
Homomorphism of GrpFP: G2b into GrpFP: G1s induced by
a |--> G1s.1 * G1s.2
b |--> G1s.1^6 * G1s.2^10
c |--> G1s.2
```
### Quotient Group Methods
A powerful technique for investigating a fp-group is to study some of its quotients. In this section we present tools for constructing various types of quotient group. Specifically, machinery is available for constructing abelian quotients, $p$-quotients, nilpotent quotients, soluble quotients, and $L(2)$, $L(3)$, and $U(3)$ quotients.
#### Abelian Quotient
The functions in this section compute information about the abelian quotient of an fp-group $G$. Some functions may require the computation of a coset table. Experienced users can control the behaviour of an implicit coset enumeration with a set of global parameters. These global parameters can be changed using the function [`SetGlobalTCParameters`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#procedure-grpfp-setglobaltcparameters). For a detailed description of the available parameters and their meanings, we refer to Subsec. [Interactive Coset Enumeration](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/subgroup-general.md#coset-enumeration).
The functions returning the abelian quotient or the abelian quotient invariants report an error if the abelian quotient cannot be computed, for example, because the relation matrix is too large. To avoid problems in user written programs or loops, the functions [`HasComputableAbelianQuotient`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-hascomputableabelianquotient) and [`HasInfiniteComputableAbelianQuotient`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-hasinfinitecomputableabelianquotient-grpfp) can be used.
##### `AbelianQuotient(G): GrpFP -> GrpAb, Map`
The maximal abelian quotient $G/G^\prime$ of the group $G$ as `GrpAb` (cf. Chapter [Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)). The natural epimorphism $\pi:G\rightarrow G/G^\prime$ is returned as second value.
##### `ElementaryAbelianQuotient(G, p): GrpFP, RngIntElt -> GrpAb, Map`
The maximal $p$-elementary abelian quotient $Q$ of the group $G$ as `GrpAb` (cf. Chapter [Abelian Groups](https://docs.magma-maths.org/FinitelyPresentedGroups/AbelianGroups/index-abelian-groups.md#chapgrpab)). The natural epimorphism $\pi:G\rightarrow Q$ is returned as second value.
##### `AbelianQuotientInvariants(G): GrpFP -> [ RngIntElt ]`
##### `AQInvariants(G): GrpFP -> [ RngIntElt ]`
Given a finitely presented group $G$, this function computes the elementary divisors of the derived quotient group $G/G'$, by constructing the relation matrix for $G$ and transforming it into Smith normal form. The algorithm
the ring of integers ${\mathbb{Z}}$ using clever heuristics to minimize the growth of coefficients.
The divisors are returned as a sequence of integers.
##### `AbelianQuotientInvariants(H): GrpFP -> [ RngIntElt ]`
##### `AQInvariants(H): GrpFP -> [ RngIntElt ]`
##### `AbelianQuotientInvariants(G, T): GrpFP, Map -> [ RngIntElt ]`
##### `AQInvariants(G, T): GrpFP, Map -> [ RngIntElt ]`
Given a subgroup $H$ of the finitely presented group $G$, this function computes the elementary divisors of the derived quotient group of $H$. (The coset table $T$ may be used to define $H$.) This is done by abelianising the Reidemeister-Schreier presentation for $H$ and then proceeding as above. The divisors are returned as a sequence of integers.
##### `AbelianQuotientInvariants(G, n): GrpFP, RngIntElt -> [ RngIntElt ]`
##### `AQInvariants(G, n): GrpFP, RngIntElt -> [ RngIntElt ]`
Given a finitely presented group $G$, this function computes the elementary divisors of the quotient group $G/N$, where $N$ is the normal subgroup of $G$ generated by the derived group $G'$ together with all $n$-th powers of elements of $G$. The algorithm constructs the relation matrix corresponding to the presentation of $G$ and computes its Smith normal form over the ring $Z/nZ$. The calculation is particularly efficient when $n$ is a small prime. The divisors are returned as a sequence of integers.
##### `AbelianQuotientInvariants(H, n): GrpFP, RngIntElt -> [ RngIntElt ]`
##### `AQInvariants(H, n): GrpFP, RngIntElt -> [ RngIntElt ]`
##### `AbelianQuotientInvariants(G, T, n): GrpFP, Map, RngIntElt -> [ RngIntElt ]`
##### `AQInvariants(G, T, n): GrpFP, Map, RngIntElt -> [ RngIntElt ]`
Given a subgroup $H$ of the finitely presented group $G$, this function computes the elementary divisors of the quotient group $H/N$, where $N$ is the normal subgroup of $H$ generated by $H'$ together with all $n$-th powers of elements of $H$. (The coset table $T$ may be used to define $H$.) This is done by abelianising the Reidemeister-Schreier presentation for $H$ and then proceeding as above. The divisors are returned as a sequence of integers.
##### `HasComputableAbelianQuotient(G): GrpFP -> BoolElt, GrpAb, Map`
Given an fp-group $G$, this function tests whether the abelian quotient of $G$ can be computed. If so, it returns the value `true`, the abelian quotient $A$ of $G$, and the natural epimorphism $\pi:G\ar A$. If the abelian quotient of $G$ cannot be computed, the value `false` is returned.
This function is especially useful to avoid runtime errors in user written loops or functions.
##### `HasInfiniteComputableAbelianQuotient(G): GrpFP -> BoolElt, GrpAb, Map`
Given an fp-group $G$, this function tests whether the abelian quotient of $G$ can be computed and is infinite. If so, it returns the value `true`, the abelian quotient $A$ of $G$, and the natural epimorphism $\pi:G\ar A$. If the abelian quotient of $G$ cannot be computed or if it is finite, the value `false` is returned.
The function first checks the modular abelian invariants for a set of small primes. If for one of these primes, the modular abelian quotient is trivial, $A$ must be finite and the function returns without actually computing the abelian quotient. If one is interested only in infinite quotients, this heuristics may save time.
##### `IsPerfect(G): GrpFP -> BoolElt`
Given an fp-group $G$, this function tries to decide whether $G$ is perfect by checking whether the abelian quotient of $G$ is trivial.
##### `TorsionFreeRank(G): GrpFP -> RngIntElt`
Given the finitely presented group $G$, return the torsion-free rank of the derived quotient group of $G$.
##### `Example: Structure of the Fibonacci Group F(7) (ex-5b646e)`
The Fibonacci group $F(7)$ has the following $2$-generator presentation:
$$
.
$$
We proceed to investigate the structure of this group.
```magma
> F := FreeGroup(2);
> F7 := quo< F | a^2*b^-2*a^-1*b^-2*(a^-1*b^-1)^2,
> a*b*a*b^2*a*b*a*b^-1*(a*b^2)^2 >;
> F7;
Finitely presented group F7 on 2 generators
Relations
a^2 * b^-2 * a^-1 * b^-2 * a^-1 * b^-1 * a^-1 * b^-1 =Id(F7)
a * b * a * b^2 * a * b * a * b^-1 * a * b^2 * a * b^2 = Id()7
```
We begin by determining the structure of the maximal abelian quotient of $F(7)$.
```magma
> AbelianQuotientInvariants(F7);
[ 29 ]
```
The maximal abelian quotient of $F(7)$ is cyclic of order 29. At this point there is no obvious way to proceed, so we attempt to determine the index of some subgroups.
```magma
> Index( F7, sub< F7 | a > );
1
```
We are in luck: $F(7)$ is generated by $a$ and so must be cyclic. This fact coupled with the knowledge that its abelian quotient has order 29 tells us that the group is cyclic of order 29.
##### `Example: Abelian Quotient Invariants Modulo 2 (ex-818c7a)`
The group $G = (8, 7\ |\ 2, 3)$ is defined by the presentation
$$
< a, b \ | \ a^8, b^7, (ab)^2, (a^{-1}b)^3 >.
$$
We consider the subgroup $H$ of $G$, generated by the words $a^2$ and $a^{-1}b$:
```magma
> G := FPGroup;
> H := sub< G | a^2, a^-1 * b >;
```
The fastest way to determine the order of the maximal 2-elementary abelian quotient of $H$ is to use the function [`AbelianQuotientInvariants`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-abelianquotientinvariants-mod):
```magma
> #AbelianQuotientInvariants(H,2);
1
```
We see that the maximal 2-elementary abelian quotient of $H$ has order $2^1$.
##### `HasFiniteAbelianQuotient(G): GrpFP -> [ RngIntElt ]`
##### `HasFiniteAQ(G): GrpFP -> [ RngIntElt ]`
Return `true` if $G$ has finite abelian quotient. The algorithm requires either a presentation for $G$, or the coset table for $G$ inside some supergroup of $G$ which has a presentation. If such a coset table cannot be computed, an error will result.
##### `AQPrimes(G): GrpFP -> [ RngIntElt ]`
Return a sequence of primes containing all prime divisors of the order of $G/G'$, if this is finite. Note that there may be primes in the return sequence which do not divide this order. If `[ 0 ]` is returned then $G/G'$ is infinite. If `[]` is returned, then $G$ is perfect. The algorithm requires either a presentation for $G$, or the coset table for $G$ inside some supergroup of $G$ which has a presentation. If such a coset table cannot be computed, an error will result.
#### $p$-Quotient
Let $F$ be a finitely presented group, $p$ a prime, and $c$ a positive integer. A $p$-quotient algorithm constructs a consistent power-conjugate presentation for the largest $p$-quotient of $F$ having lower exponent-$p$ class at most $c$. The $p$-quotient algorithm used by Magma is part of the ANU $p$-Quotient program. For details of the algorithm, see [[NO96](https://docs.magma-maths.org/references.md#cite-newman-obrien)]. In Magma the result is returned as a group of type `GrpPC` (cf. Chapter [Finite Soluble Groups](https://docs.magma-maths.org/FiniteGroups/FiniteSolubleGroups/index-finite-soluble-groups.md#chapgrppc)).
Assume that the $p$-quotient has order $p^n$, Frattini rank $d$, and that its generators are $a_1, \ldots, a_n$. Then the power-conjugate presentation constructed has the following additional structure. The set $\{a_1,\ldots,a_d\}$ is a generating set for $G$. For each $a_k$ in $\{a_{d+1},\ldots,a_n\}$, there is at least one relation whose right hand side is $a_k$. One of these relations is taken as the *definition* of $a_k$. (The list of definitions is also returned by `pQuotient`.) The power-conjugate generators also have a *weight* associated with them: a generator is assigned a weight corresponding to the stage at which it is added and this weight is extended to all normal words in a natural way.
The $p$-quotient function and its associated commands allows the user to construct a power-conjugate presentation (pcp) for a $p$-group. Note that there is also a process version of the $p$-quotient algorithm, which gives the user complete control over its execution. For a description, we refer to Subsec. [p-Quotient Process](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#sectpquotientprocess).
##### `pQuotient(F, p, c: parameters): GrpFP, RngIntElt, RngIntElt -> GrpPC, Map, SeqEnum, BoolElt`
Given an fp-group $F$, a prime $p$, and a positive integer $c$, construct a pcp for the largest $p$-quotient $G$ of $F$ having lower exponent-$p$ class at most $c$. If $c$ is given as $0$, then the limit 127 is placed on the class. The prime $p$ must be less than $65536$. The function also returns the natural homomorphism $\pi$ from $F$ to $G$, a sequence $S$ describing the definitions of the pc-generators of $G$ and a flag indicating whether $G$ is the maximal $p$-quotient of $F$.
The $k$-th element of $S$ is a sequence of two integers, describing the definition of the $k$-th pc-generator $G.k$ of $G$ as follows.
- If $S[k] = [0,r]$, then $G.k$ is defined via the image of $F.r$ under $\pi$.
- If $S[k] = [r,0]$, then $G.k$ is defined via the power relation for $G.r$.
- If $S[k] = [r,s]$, then $G.k$ is defined via the conjugate relation involving $G.r^{G.s}$.
There exist a number of parameters for controlling the behaviour of this function, which are described below.
##### `Example: Largest 2-Quotient of Class 6 (ex-e7285c)`
We construct the largest $2$-quotient of class $6$ for a two-generator, two-relator group.
```magma
> F := FreeGroup(2);
> G := quo< F | (b, a, a) = 1, (a * b * a)^4 = 1 >;
> Q, fQ := pQuotient(G, 2, 6);
> Order(Q);
524288
> fQ;
Mapping from: GrpFP: G to GrpPC: Q
```
The parameters available for the function [`pQuotient`](https://docs.magma-maths.org/FinitelyPresentedGroups/FPGroups/quotient.md#function-grpfp-1-pquotient) are:
```magma
Exponent: RngIntElt Default: 0
```
If `Exponent := m`, enforce the exponent law, $x^m = 1$, on the group.
```magma
Metabelian: BoolElt Default: false
```
If `Metabelian := true`, then a consistent pcp is constructed for the largest metabelian $p$-quotient of $F$ having lower exponent-$p$ class at most $c$.
```magma
Print: RngIntElt Default: 0
```
This parameter controls the volume of printing. By default its value is that returned by `GetVerbose("pQuotient")`, which is $0$ unless it has been changed through use of `SetVerbose`. The effect is the following:
`Print := 0`: No output.
`Print := 1`: Report order of $p$-quotient at each class.
`Print := 2`: Report statistics and redundancy information about tails, consistency, collection of relations, and exponent enforcement components of calculation.
`Print := 3`: Report in detail on the construction of each class.
Note that the presentation displayed is a *power-commutator* presentation (since this is the version stored by the $p$-quotient).
```magma
Workspace: RngIntElt Default: 5000000
```
The amount of space requested for the $p$-quotient computation.
##### `Example: Largest 3-Quotient of Class 6 (ex-090d87)`
We construct the largest $3$-quotient of class $6$ for a two-generator group of exponent $9$.
```magma
> F := FreeGroup(2);
> G := quo< F | a^3 = b^3 = 1 >;
> q := pQuotient(G, 3, 6: Print := 1, Exponent := 9);
Lower exponent-3 central series for G
Group: G to lower exponent-3 central class 1 has order 3^2
Group: G to lower exponent-3 central class 2 has order 3^3
Group: G to lower exponent-3 central class 3 has order 3^5
Group: G to lower exponent-3 central class 4 has order 3^7
Group: G to lower exponent-3 central class 5 has order 3^9
Group: G to lower exponent-3 central class 6 has order 3^11
```
##### `Example: Metabelian 5-Quotient (ex-bccae5)`
We use the metabelian parameter to construct a metabelian $5$-quotient of the group
$$
< a, b\ |\ a^{625} = b^{625} = 1, (b, a, b) = 1, (b, a, a, a, a) = (b, a)^5>.
$$
```magma
> F := FreeGroup(2);
> G := quo< F | a^625 = b^625 = 1, (b, a, b) = 1,
> (b, a, a, a, a) = (b, a)^5 >;
> q := pQuotient(G, 5, 20: Print := 1, Metabelian := true);
Lower exponent-5 central series for G
Group: G to lower exponent-5 central class 1 has order 5^2
Group: G to lower exponent-5 central class 2 has order 5^5
Group: G to lower exponent-5 central class 3 has order 5^8
Group: G to lower exponent-5 central class 4 has order 5^11
Group: G to lower exponent-5 central class 5 has order 5^12
Group: G to lower exponent-5 central class 6 has order 5^13
Group: G to lower exponent-5 central class 7 has order 5^14
Group: G to lower exponent-5 central class 8 has order 5^15
Group: G to lower exponent-5 central class 9 has order 5^16
Group: G to lower exponent-5 central class 10 has order 5^17
Group: G to lower exponent-5 central class 11 has order 5^18
Group: G to lower exponent-5 central class 12 has order 5^19
Group: G to lower exponent-5 central class 13 has order 5^20
Group completed. Lower exponent-5 central class = 13, order = 5^20
```
##### `Example: Largest 2-Generator Group of Exponent 5 (ex-a5910e)`
In the final example, we construct the largest finite 2-generator group having exponent $5$.
```magma
> F := FreeGroup(2);
> q := pQuotient (F, 5, 14: Print := 1, Exponent := 5);
Lower exponent-5 central series for F
Group: F to lower exponent-5 central class 1 has order 5^2
Group: F to lower exponent-5 central class 2 has order 5^3
Group: F to lower exponent-5 central class 3 has order 5^5
Group: F to lower exponent-5 central class 4 has order 5^8
Group: F to lower exponent-5 central class 5 has order 5^10
Group: F to lower exponent-5 central class 6 has order 5^14
Group: F to lower exponent-5 central class 7 has order 5^18
Group: F to lower exponent-5 central class 8 has order 5^22
Group: F to lower exponent-5 central class 9 has order 5^28
Group: F to lower exponent-5 central class 10 has order 5^31
Group: F to lower exponent-5 central class 11 has order 5^33
Group: F to lower exponent-5 central class 12 has order 5^34
Group completed. Lower exponent-5 central class = 12, order = 5^34
```
##### `HaspQuotientDefinitions(G): GrpPC -> BoolElt`
Given a pc-group $G$, this function returns `true` iff and only if $G$ has generator definitions set by `pQuotient`.
##### `pQuotientDefinitions(G): GrpPC -> SeqEnum`
Given a pc-group G, this function returns the generator definitions set by `pQuotient`, and will throw an error if none exist.
#### $p$-Quotient Process
In this and the next subsections the interative $p$-quotient process is described.
##### `pQuotientProcess(F, p, c: parameters): GrpFP, RngIntElt, RngIntElt -> Process`
Given an fp-group $F$, a prime $p$, and a positive integer $c$, create a $p$-quotient process for the group $F$ with the indicated arguments. As part of the initialisation of the process, a pcp for the largest $p$-quotient of $F$ having class at most $c$ will be constructed. If $c$ is given as $0$, then the limit 127 is placed on the class. This function supports the same parameters as `pQuotient` and returns a process $P$.
##### `NextClass(~P : parameters): GrpPCpQuotientProc`
##### `NextClass(~P, k : parameters): GrpPCpQuotientProc, RngIntElt`
```magma
Exponent : RngIntElt Default:
Metabelian : BoolElt Default:
Print : RngIntElt Default:
MaxOccurrence: [ RngIntElt ] Default: []
```
Assumes that a pcp has already been constructed for the class $c$ quotient of $F$. It seeks to construct a pcp for the class $c + 1$ $p$-quotient of $F$. If $k$ is supplied, continue to construct until the pcp for the largest quotient of class $k$ is constructed.
The parameters `Exponent`, `Print`, and `Metabelian` are used as before. If `MaxOccurrence := Q`, then the sequence $Q$ has length equal to the rank of the class 1 quotient of $F$; its entries are integers which specify the maximum number of occurrences of the class 1 generators in the definitions of pcp generators of $F$. An entry of $0$ for a particular generator indicates that no limit is placed on the number of occurrences of this generator.
Care should be exercised when supplying values for parameters. Once set, they retain their values until explicitly reassigned.
##### Using $p$-Quotient Interactively
We assume that we have constructed a pcp for the largest class $c$ $p$-quotient of $F$ and now seek to construct a pcp for the largest class $c + 1$ $p$-quotient. The following options allow the user to construct a pcp for the next class of the group interactively. The steps are laid out in one of a number of natural sequences in which they may be executed. Some of them may be interleaved; however, the user should pay particular attention to the assumptions mentioned below. The procedures that drive the process do not verify that the assumptions are satisfied.
###### `StartNewClass(~P: parameters): GrpPCpQuotientProc`
If $P$ is a process for a class $c$ $p$-quotient, commence construction of class $c + 1$.
###### `Tails(~P: parameters): GrpPCpQuotientProc`
###### `Tails(~P, k: parameters): GrpPCpQuotientProc, RngIntElt`
```magma
Metabelian: BoolElt Default: false
```
Add tails to the current pcp; default is to add all tails for this class. If $k$ is supplied, then tails for weight $k$ only are added; in this case, it is assumed that the tails for each of weight $c + 1, c, \ldots, k + 1$ have already been added. The valid range of $k$ is $2, \ldots, c + 1$. The one valid parameter is `Metabelian`; if `true`, then only the tails for the metabelian $p$-quotient are inserted.
###### `Consistency(~P: parameters): GrpPCpQuotientProc`
###### `Consistency(~P, k: parameters): GrpPCpQuotientProc`
```magma
Metabelian: BoolElt Default: false
```
Apply the consistency algorithm to the pcp to compute any redundancies among the tails already added. Default is to apply it to all tails; in this case, it is assumed that all tails have been added. If $k$ is supplied, it is assumed that tails for weight $k$ have been added; in this case, the tails added for weight $k$ only are checked. The range of $k$ is $3, \ldots, c + 1$. The one valid parameter is `Metabelian`; if `true`, we assume that the tails inserted were those for a metabelian $p$-quotient and hence invoke the (less expensive) metabelian consistency algorithm.
###### `CollectRelations(~P): GrpPCpQuotientProc`
Collect the defining relations (if any) in the current pcp. If the tails operation is not complete, then the relations may be evaluated incorrectly.
###### `ExponentLaw(~P : parameters): GrpPCpQuotientProc`
###### `ExponentLaw(~P, Start, Fin: parameters): GrpPCpQuotientProc, RngIntElt, RngIntElt`
```magma
Exponent : RngIntElt Default: 0
Print : RngIntElt Default: 1
Trial : BoolElt Default: false
ShortList : BoolElt Default: false
DisplayList : BoolElt Default: false
IdentifyFilters: BoolElt Default: false
InitialSegment : [] Default: []
Report : RngIntElt Default: 0
```
Enforce the supplied exponent law on the current pcp. If `Start` and `Fin` are supplied, then enforce the law for those weights between `Start` and `Fin`; otherwise, enforce the law for all weights. It is assumed that the tails operation is complete. If the display parameter `DisplayLevel` (which may be set using `SetDisplayLevel`) has value $2$, those words whose powers are collected and give redundancies among the pcp generators are printed out. If `DisplayLevel` has value $3$, all words whose powers are collected are printed out. The following additional parameters are available:
If `Exponent := m`, enforce the exponent law, $x^m = 1$, on the group.
As for `pQuotient`.
Generate the list of words used to enforce the exponent law and print out statistics but do not power words or echelonise the results.
Generate the list of enforcement words whose entries have the form $w$ or $1 \ast w$ where $w$ is an element of the Frattini subgroup of $F$.
Display the list of all enforcement words generated – not just those which are collected.
Identify filters used to eliminate words from list.
If `InitialSegment := w`, generate only those enforcement words which have $w$ as an initial segment, where $w$ is supplied as a sequence of generator-exponent pairs.
If `Report := n`, report after computing the powers of each collection of $n$ enforcement words.
###### `EliminateRedundancy(~P): GrpPCpQuotientProc`
Eliminate all redundant generators from the pcp defined by process $P$. This operation may be performed at any time.
We now list the remaining functions which can be applied to a $p$Quotient process.
###### `Display(P): GrpPCpQuotientProc`
###### `Display(P, DisplayLevel): GrpPCpQuotientProc, RngIntElt`
Display the pcp for the $p$-quotient $G$ of the fp-group $F$. The argument `DisplayLevel` may be 1, 2, or 3, and is used to control the amount of information given:
**`1` :**
Display order and class of $G$;
**`2` :**
Display non-trivial relations for $G$;
**`3` :**
Display the structure of pcp generators of $G$, non-trivial relations of $G$, and the map from the defining generators of $F$ to the pcp generators of $G$.
The presentation displayed by this function is in power-commutator form. If `DisplayLevel` is not supplied, the information displayed is determined by its existing (or default) value.
###### `RevertClass(~P): GrpPCpQuotientProc`
Given a pcp for the class $c + 1$ $p$-quotient of $F$, this procedure reverts to the pcp for the class $c$ $p$-quotient of $F$. Note that this command can be applied only **once** during construction of a single class.
###### `pCoveringGroup(~P): GrpPCpQuotientProc`
###### `pCoveringGroup(G): GrpPC -> GrpPC`
Given a process or a pcp for a $p$-group, this procedure computes a pcp for the $p$-covering group of this group. In the process case, it is equivalent to `Tails(~P); Consistency(~P); EliminateRedundancy(~P)`.
###### `GeneratorStructure(P): GrpPCpQuotientProc`
###### `GeneratorStructure(P, Start, Fin): GrpPCpQuotientProc, RngIntElt, RngIntElt`
Display the structure of the generators in the pcp. If *Start* and *Fin* are given, then print out the structure of those pcp generators numbered from *Start* to *Fin*.
###### `Jacobi(~P, c, b, a, ~r): GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt -> RngIntElt ->`
###### `Jacobi(~P, c, b, a): GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt -> RngIntElt ->`
Calculate the Jacobi $c, b, a$ and echelonise the resulting relation against the current pcp. If a redundant generator results from the echelonisation, the optional variable $r$ is the number of that generator; otherwise $r$ has value $0$.
###### `Collect(P, Q): GrpPCpQuotientProc, [ ] -> [ RngIntElt ] ->`
The sequence $Q$, consisting of generator-exponent pairs, defines a word $w$ in the pcp generators of the group defined by the process $P$. Collect this word and return the resulting normal word as an exponent vector.
###### `EcheloniseWord(~P, ~r): GrpPCpQuotientProc -> RngIntElt`
###### `EcheloniseWord(~P): GrpPCpQuotientProc -> RngIntElt`
Echelonise the word most recently collected using `Collect` against the relations of the pcp. If a redundant generator results from the echelonisation, the optional variable $r$ is the number of that generator; otherwise $r$ has value $0$. This function must be called immediately after `Collect`.
###### `SetDisplayLevel(~P, Level): GrpPCpQuotientProc, RngIntElt`
This procedure alters the display level for the process to the supplied value, `Level`.
###### `ExtractGroup(P): GrpPCpQuotientProc -> GrpPC`
Extract the group $G$ defined by the pcp associated with the process $P$, as a member of the category `GrpPC` of finite soluble groups. The function also returns the natural homomorphism $\pi$ from the original group $F$ to $G$, a sequence $S$ describing the definitions of the pc-generators of $G$ and a flag indicating whether $G$ is the maximal $p$-quotient of $F$.
The $k$-th element of $S$ is a sequence of two integers, describing the definition of the $k$-th pc-generator $G.k$ of $G$ as follows.
- If $S[k] = [0,r]$, then $G.k$ is defined via the image of $F.r$ under $\pi$.
- If $S[k] = [r,0]$, then $G.k$ is defined via the power relation for $G.r$.
- If $S[k] = [r,s]$, then $G.k$ is defined via the conjugate relation involving $G.r^{G.s}$.
###### `Order(P): GrpPCpQuotientProc -> RngIntElt`
The order of the group defined by the pcp associated with the process $P$.
###### `FactoredOrder(P): GrpPCpQuotientProc -> [ ]`
The factored order of the group defined by the pcp associated with the process $P$.