Finitely Presented Groups
- Introduction
- Construction of an FP-Group
- Introduction
- Quotient Group Constructor
- The FP-Group Constructor
- Accessing the Defining Generators and Relations
- Operations on Words
Eliminate(u, x, v): GrpFPElt, GrpFPElt, GrpFPElt → GrpFPEltEliminate(U, x, v): { GrpFPElt }, GrpFPElt, GrpFPElt → { GrpFPElt }Match(u, v, f): GrpFPElt, GrpFPElt, RngIntElt → BoolElt, RngIntEltRotateWord(u, n): GrpFPElt, RngIntElt → GrpFPEltSubstitute(u, f, n, v): GrpFPElt, RngIntElt, RngIntElt, GrpFPElt → GrpFPEltSubword(u, f, n): GrpFPElt, RngIntElt, RngIntElt → GrpFPElt- Example: Operations on Words in a Free Group
- Operations on Presentations
AddGenerator(G): GrpFP → GrpFPAddGenerator(G, w): GrpFP, GrpFPElt → GrpFPAddRelation(G, r): GrpFP, RelElt → GrpFPAddRelation(G, g): GrpFP, GrpFPElt → GrpFPAddRelation(G, r, i): GrpFP, RelElt, RngIntElt → GrpFPAddRelation(G, g, i): GrpFP, GrpFPElt, RngIntElt → GrpFPDeleteGenerator(G, x): GrpFP, GrpFPElt → GrpFPDeleteRelation(G, r): GrpFP, RelElt → GrpFPDeleteRelation(G, g): GrpFP, GrpFPElt → GrpFPDeleteRelation(G, i): GrpFP, RngIntElt → GrpFPReplaceRelation(G, s, r): GrpFP, RelElt, RelElt → GrpFPReplaceRelation(G, h, r): GrpFP, GrpFPElt, RelElt → GrpFPReplaceRelation(G, s, g): GrpFP, RelElt, GrpFPElt → GrpFPReplaceRelation(G, h, g): GrpFP, GrpFPElt, GrpFPElt → GrpFPReplaceRelation(G, i, r): GrpFP, RngIntElt, RelElt → GrpFPReplaceRelation(G, i, g): GrpFP, RngIntElt, GrpFPElt → GrpFP- Example: Replacing a Relation
- Simplification
- Standard Constructions
- Familiar Groups as FP-Groups
- Construction of Extensions
- Conversion to FP-Groups
FPGroup(G): GrpPerm → GrpFP, Hom(Grp)FPGroup(G): GrpMat → GrpFP, Hom(Grp)- Example: FP-Group from a Permutation Group
FPGroupStrong(G): GrpPerm → GrpFP, Hom(Grp)FPGroupStrong(G): GrpMat → GrpFP, Hom(Grp)FPGroupStrong(G, N): GrpPerm, GrpPerm → GrpFP, Hom(Grp)- Example: FP-Group on Strong Generators
FPGroup(G): GrpPC → GrpFP, Hom(Grp)FPGroup(G): GrpGPC → GrpFP, Hom(Grp)FPGroup(G): GrpAb → GrpFP, Hom(Grp)- Example: Converting a Polycyclic Group to an FP-Group
CoxeterFPGroup(W): GrpFPCox → GrpFP, MapCoxeterFPGroup(W): GrpPermCox → GrpFP, Map- Example: Coxeter Group as an FP-Group
- Properties of an FP-Group
- Subgroups
- Specification of a Subgroup
- The Todd-Coxeter Algorithm
ToddCoxeter(G, H: parameters): GrpFP, GrpFP → RngIntElt, Map, RngIntElt, RngIntEltIndex(G, H: parameters): GrpFP, GrpFP → RngIntEltFactoredIndex(G, H: parameters): GrpFP, GrpFP → [ <RngIntElt, RngIntElt> ]- Example: Todd-Coxeter Enumeration of 448 Cosets
- Example: Todd-Coxeter for the Harada-Norton Group
- Example: Orders of a Parametrised Family of Groups
- Interactive Coset Enumeration
- Implicit Invocation of the Todd-Coxeter Algorithm
- Coset Spaces and Tables
- Coset Tables
- Coset Spaces: Induced Homomorphism
- Coset Spaces: Construction
- Coset Spaces: Elementary Operations
- Accessing Information
# V: GrpFPCos → RngIntEltAction(V): GrpFPCos → Map<i, w> @ T: GrpFPCosElt, GrpFPElt, Map → GrpFPEltT(i, w): Map, GrpFPCosElt, GrpFPElt → GrpFPEltExplicitCoset(V, i): GrpFPCos, RngIntElt → GrpFPCosEltIndexedCoset(V, w): GrpFPCos, GrpFPElt → GrpFPCosEltIndexedCoset(V, C): GrpFPCos, GrpFPCosElt → GrpFPCosEltGroup(V): GrpFPCos → GrpFPSubgroup(V): GrpFPCos → GrpFPIsComplete(V): GrpFPCos → BoolEltExcludedConjugates(V): GrpFPCos → { GrpFPElt }ExcludedConjugates(T): Map → { GrpFPElt }Transversal(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, MapRightTransversal(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, Map- Example: Accessing a Coset Table
- Example: Working with a Coset Space
- Example: Derived Subgroup via Coset Enumeration
- Example: Adding Excluded Conjugates
- Double Coset Spaces: Construction
DoubleCoset(G, H, g, K ): GrpFP, GrpFP, GrpFPElt, GrpFP → GrpFPDcosEltDoubleCosets(G, H, K): GrpFP, GrpFP, GrpFP → { GrpFPDcosElt }- Example: Double Cosets in the Infinite Dihedral Group
- Coset Spaces: Selection of Cosets
- Constructing a Presentation for a Subgroup
- Subgroups of Finite Index
- Low Index Subgroups
LowIndexSubgroups(G, R : parameters): GrpFP, RngIntElt → [ GrpFP ]LowIndexSubgroups(G, R : parameters): GrpFP, RngIntElt → [ { GrpFPElt } ]LowIndexSubgroups(G, R: parameters): GrpFP, <RngIntElt, RngIntElt> → [ GrpFP ]LowIndexSubgroups(G, R: parameters): GrpFP, <RngIntElt, RngIntElt> → [ { GrpFPElt } ]- Example: Tutte's 8-Cage and the Conder Graph
- Example: Conjugacy Classes via Low Index Subgroups
LowIndexProcess(G, R : parameters): GrpFP, RngIntElt → Process(Lix)LowIndexProcess(G, R: parameters): GrpFP, < RngIntElt, RngIntElt > → Process(Lix)NextSubgroup(~P): GrpFPLixProcNextSubgroup(~P, ~G): GrpFPLixProc, GrpFPExtractGroup(P): GrpFPLixProc → GrpFPExtractGenerators(P): GrpFPLixProc → { GrpFPElt }IsEmpty(P): GrpFPLixProc → BoolEltIsValid(P): GrpFPLixProc → BoolElt- Example: Low Index Subgroups of a Triangle Group
- Example: PSL(2,8) as a Quotient of a Triangle Group
- Example: Proving a Group Is Infinite
LowIndexNormalSubgroups(G, n: parameters): GrpFP, RngIntElt → [ Rec ]
- Operations for Subgroups of Finite Index
H ^ u: GrpFP, GrpFPElt → GrpFPConjugate(H, u): GrpFP, GrpFPElt → GrpFPH meet K: GrpFP, GrpFP → GrpFPCore(G, H): GrpFP, GrpFP → GrpFPGeneratingWords(G, H): GrpFP, GrpFP → { GrpFPElt }MaximalOvergroup(G, H): GrpFP, GrpFP → GrpFPMinimalOvergroup(G, H): GrpFP, GrpFP → GrpFPH ^ G: GrpFP, GrpFP → GrpFPNormalClosure(G, H): GrpFP, GrpFP → GrpFPNormaliser(G, H): GrpFP, GrpFP → GrpFPNormalizer(G, H): GrpFP, GrpFP → GrpFPSchreierGenerators(G, H : parameters): GrpFP, GrpFP → { GrpFPElt }SchreierSystem(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, MapTransversal(G, H): GrpFP, GrpFP → {@ GrpFPElt @}, MapTransversal(G, H, K): GrpFP, GrpFP, GrpFP → {@ GrpFPElt @}, Map- Example: Subgroup Constructions
- Example: Schreier Generators
- Properties of Subgroups
u in H: GrpFPElt, GrpFP → BoolEltu notin H: GrpFPElt, GrpFP → BoolEltH eq K: GrpFP, GrpFP → BoolEltH ne K: GrpFP, GrpFP → BoolEltH subset K: GrpFP, GrpFP → BoolEltIsSubgroup(H,K): GrpFP, GrpFP → BoolEltH notsubset K: GrpFP, GrpFP → BoolEltIsConjugate(G, H, K): GrpFP, GrpFP, GrpFP → BoolElt, GrpFPEltIsNormal(G, H): GrpFP, GrpFP → BoolEltIsMaximal(G, H): GrpFP, GrpFP → BoolEltIsSelfNormalizing(G, H): GrpFP, GrpFP → BoolElt- Example: Subgroup Predicates in a Space Group
- Example: Maximal Subgroups of the Hall-Janko Group
- Finite FP-Groups
- Homomorphisms
- General Remarks
- Construction of Homomorphisms
- Accessing Homomorphisms
- Constructing Homomorphisms onto Finite Groups
Homomorphisms(F, G, A : parameters): GrpFP, GrpPerm, GrpPerm → [ HomGrp ]Homomorphisms(F, G : parameters): GrpFP, GrpPerm → [ HomGrp ]- Example: Finding Homomorphisms onto Finite Groups
Homomorphisms(F, G, A : parameters): GrpFP, GrpPC, GrpPC → [ HomGrp ]Homomorphisms(F, G : parameters): GrpFP, GrpPC → [ HomGrp ]HomomorphismsProcess(F, G, A : parameters): GrpFP, GrpPerm, GrpPerm → GrpFPHomsProcHomomorphismsProcess(F, G : parameters): GrpFP, GrpPerm → GrpFPHomsProcNextElement(~P): GrpFPHomsProcIsEmpty(P): GrpFPHomsProc → BoolEltIsValid(P): GrpFPHomsProc → BoolEltDefinesHomomorphism(P): GrpFPHomsProc → BoolEltHomomorphism(P): GrpFPHomsProc → HomGrp# P: GrpFPHomsProc → RngIntElt- Example: Braid Group onto PSL(2, 16)
- Example: Proving Infiniteness Interactively
- Searching for Isomorphisms
- Quotient Group Methods
- Abelian Quotient
AbelianQuotient(G): GrpFP → GrpAb, MapElementaryAbelianQuotient(G, p): GrpFP, RngIntElt → GrpAb, MapAbelianQuotientInvariants(G): GrpFP → [ RngIntElt ]AQInvariants(G): GrpFP → [ RngIntElt ]AbelianQuotientInvariants(H): GrpFP → [ RngIntElt ]AQInvariants(H): GrpFP → [ RngIntElt ]AbelianQuotientInvariants(G, T): GrpFP, Map → [ RngIntElt ]AQInvariants(G, T): GrpFP, Map → [ RngIntElt ]AbelianQuotientInvariants(G, n): GrpFP, RngIntElt → [ RngIntElt ]AQInvariants(G, n): GrpFP, RngIntElt → [ RngIntElt ]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 ]HasComputableAbelianQuotient(G): GrpFP → BoolElt, GrpAb, MapHasInfiniteComputableAbelianQuotient(G): GrpFP → BoolElt, GrpAb, MapIsPerfect(G): GrpFP → BoolEltTorsionFreeRank(G): GrpFP → RngIntElt- Example: Structure of the Fibonacci Group F(7)
- Example: Abelian Quotient Invariants Modulo 2
HasFiniteAbelianQuotient(G): GrpFP → [ RngIntElt ]HasFiniteAQ(G): GrpFP → [ RngIntElt ]AQPrimes(G): GrpFP → [ RngIntElt ]
- \(p\)-Quotient
- \(p\)-Quotient Process
pQuotientProcess(F, p, c: parameters): GrpFP, RngIntElt, RngIntElt → ProcessNextClass(~P : parameters): GrpPCpQuotientProcNextClass(~P, k : parameters): GrpPCpQuotientProc, RngIntElt- Using \(p\)-Quotient Interactively
StartNewClass(~P: parameters): GrpPCpQuotientProcTails(~P: parameters): GrpPCpQuotientProcTails(~P, k: parameters): GrpPCpQuotientProc, RngIntEltConsistency(~P: parameters): GrpPCpQuotientProcConsistency(~P, k: parameters): GrpPCpQuotientProcCollectRelations(~P): GrpPCpQuotientProcExponentLaw(~P : parameters): GrpPCpQuotientProcExponentLaw(~P, Start, Fin: parameters): GrpPCpQuotientProc, RngIntElt, RngIntEltEliminateRedundancy(~P): GrpPCpQuotientProcDisplay(P): GrpPCpQuotientProcDisplay(P, DisplayLevel): GrpPCpQuotientProc, RngIntEltRevertClass(~P): GrpPCpQuotientProcpCoveringGroup(~P): GrpPCpQuotientProcpCoveringGroup(G): GrpPC → GrpPCGeneratorStructure(P): GrpPCpQuotientProcGeneratorStructure(P, Start, Fin): GrpPCpQuotientProc, RngIntElt, RngIntEltJacobi(~P, c, b, a, ~r): GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt → RngIntElt ->Jacobi(~P, c, b, a): GrpPCpQuotientProc, RngIntElt, RngIntElt, RngIntElt → RngIntElt ->Collect(P, Q): GrpPCpQuotientProc, [ <RngIntElt, RngIntElt> ] → [ RngIntElt ] ->EcheloniseWord(~P, ~r): GrpPCpQuotientProc → RngIntEltEcheloniseWord(~P): GrpPCpQuotientProc → RngIntEltSetDisplayLevel(~P, Level): GrpPCpQuotientProc, RngIntEltExtractGroup(P): GrpPCpQuotientProc → GrpPCOrder(P): GrpPCpQuotientProc → RngIntEltFactoredOrder(P): GrpPCpQuotientProc → [ <RngIntElt, RngIntElt> ]NumberOfPCGenerators(P): GrpPCpQuotientProc → RngIntEltpClass(P): GrpPCpQuotientProc → RngIntEltNuclearRank(G): GrpPC → RngIntEltNuclearRank(P): GrpPCpQuotientProc → RngIntEltpMultiplicatorRank(G): GrpPC → RngIntEltpMultiplicatorRank(P): GrpPCpQuotientProc → RngIntElt- Example: Extending a 3-Quotient Interactively
- Example: Bounding Definitions of Pcp Generators
- Example: Class 7 Quotient of R(2, 5)
- Example: Interactively Extending a Class 6 Quotient
- Nilpotent Quotient
- Soluble Quotient
SolvableQuotient(G : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolubleQuotient(G : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgElt- Example: Soluble Quotient of a Two-Generator Group
SolvableQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolubleQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolvableQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgEltSolubleQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgElt- Example: Order of a Soluble Quotient
- Soluble Quotient Advanced
- Introduction
- Construction
- Calculating the Relevant Primes
- The Intrinsics
SolubleQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolvableQuotient(F, n : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolubleQuotient(F : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolvableQuotient(F : parameters): GrpFP, RngIntElt → GrpPC, Map, SeqEnum, MonStgEltSolubleQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgEltSolvableQuotient(F, P : parameters): GrpFP, Set → GrpPC, Map, SeqEnum, MonStgElt
- Simple Group Quotients
SimpleQuotients(F, deg1, deg2, ord1, ord2: parameters): GrpFP, RngIntElt, RngIntElt, RngIntElt, RngIntElt → ListSimpleQuotients(F, ord1, ord2: parameters): GrpFP, RngIntElt, RngIntElt → ListSimpleQuotients(F, ord2: parameters): GrpFP, RngIntElt → ListSimpleQuotientProcess(F, deg1, deg2, ord1, ord2: parameters): GrpFP, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RecNextSimpleQuotient(~P): RecIsEmptySimpleQuotientProcess(P): Rec → BoolEltSimpleEpimorphisms(P): Rec → SeqEnum, Tup- Example: Simple Quotients of a Perfect Group
- The \({\rm L}_2\)-Quotient Algorithm
- Infinite L2 Quotients
- The \({\rm L}_3{\rm U}_3\)-Quotient Algorithm
- \(KG\)-Modules
GModulePrimes(G, A): GrpFP, GrpFP → SetMultiGModulePrimes(G, A, B): GrpFP, GrpFP, GrpFP → SetMultiGModule(G, A, p): GrpFP, GrpFP, RngIntElt → ModGrp, MapGModule(G, A, B, p): GrpFP, GrpFP, GrpFP, RngIntElt → ModGrp, MapGModule(G, A, B): GrpFP, GrpFP, GrpFP → ModGrp, MapPullback(f, N): Map, ModGrp → GrpFP- Example: KG-Modules of a Finitely Presented Group
- Example: Relevant Primes for Module Construction
- Some Developed Examples
- Bibliography