Abelian Groups

• Introduction
• Construction of a Finitely Presented Abelian Group and its Elements
  • The Free Abelian Group
    • FreeAbelianGroup(n): RngIntElt → GrpAb
    • Example: Free Abelian Group
  • Relations
    • w₁ = w₂: GrpAbElt, GrpAbElt → Rel
    • r[1]: GrpAbRel, RngIntElt → GrpAbElt
    • LHS(r): Rel → GrpAbElt
    • r[2]: GrpAbRel, RngIntElt → GrpAbElt
    • RHS(r): Rel → GrpAbElt
    • Parent(r): RelElt → GrpAb
    • Example: Relations
  • Specification of a Presentation
    • AbelianGroup< X | R >: List(Var), List(GrpAbRel) → GrpAb, Hom(GrpAb)
    • Example: Abelian Group
    • AbelianGroup([n₁,...,nᵣ]): [ RngIntElt ] → GrpAb
    • Example: Abelian Group2
  • Accessing the Defining Generators and Relations
    • A . i: GrpAb, RngIntElt → GrpAbElt
    • Generators(A): GrpAb → { GrpAbElt}
    • NumberOfGenerators(A): GrpAb → RngIntElt
    • Ngens(A): GrpAb → RngIntElt
    • Parent(u): GrpAbElt → GrpAb
    • Relations(A): GrpAb → [ Rel ]
    • RelationMatrix(A): GrpAb → Mtrx
• Construction of a Generic Abelian Group
  • Specification of a Generic Abelian Group
    • GenericAbelianGroup(U: parameters): . → GrpAbGen
    • Example: Creation
  • Accessing Generators
    • Universe(A): GrpAbGen
    • A . i: GrpAbGen, RngIntElt → GrpAbGenElt
    • Generators(A): GrpAbGen → [ GrpAbGenElt ]
    • UserGenerators(A): GrpAbGen → [ GrpAbGenElt ]
    • NumberOfGenerators(A): GrpAbGen → RngIntElt
    • Ngens(A): GrpAbGen → RngIntElt
  • Computing Abelian Group Structure
    • AbelianGroup(A: parameters): GrpAbGen → GrpAb, Map
    • Example: Group Computation
• Elements
  • Construction of Elements
    • A ! [a₁, ... ,aₙ]: GrpAb, [RngIntElt] → GrpAbElt
    • A ! [a₁, ... ,aₙ]: GrpAbGen, [RngIntElt] → GrpAbGenElt
    • A ! e: GrpAbGen, Elt → GrpAbGenElt
    • A ! g: GrpAbGen, GrpAbGenElt → GrpAbGenElt
    • A ! n: GrpAb, RngIntElt → GrpAbElt
    • Random(A): GrpAbGen → GrpAbGenElt
    • Identity(A): GrpAb → GrpAbElt
    • Id(A): GrpAb → GrpAbElt
    • A ! 0: GrpAb, RngIntElt → GrpAbElt
  • Representation of an Element
    • Representation(g): GrpAbGenElt → [RngIntElt]
    • ElementToSequence(g): GrpAbGenElt → [RngIntElt]
    • Eltseq(g): GrpAbGenElt → [RngIntElt]
    • UserRepresentation(g): GrpAbGenElt → [RngIntElt]
    • Representation(S, g): SeqEnum, GrpAbGenElt → [RngIntElt], RngIntElt
    • Example: Element Creation And Rep
  • Arithmetic with Elements
    • u + v: GrpAbElt, GrpAbElt → GrpAbElt
    • - u: GrpAbElt → GrpAbElt
    • u - v: GrpAbElt, GrpAbElt → GrpAbElt
    • m * u: RngIntElt, GrpAbElt → GrpAbElt
    • u * m: GrpAb, RngIntElt → GrpAbElt
• Construction of Subgroups and Quotient Groups
  • Construction of Subgroups
    • sub<A | L>: GrpAb, List → GrpAb, Map
    • Example: Subgroup Creation
    • sub<A | L: parameters>: GrpAbGen, List → GrpAbGen
    • Example: Generic Subgroup Creation
  • Construction of Quotient Groups
    • quo<F | R>: GrpAb, List → GrpAb, Hom(GrpAb)
    • A / B: GrpAb, GrpAb → GrpAb
• Standard Constructions and Conversions
  • AbelianGroup(GrpAb, Q): Cat, [ RngIntElt ] → GrpAb
  • AbelianGroup(Q): [ RngIntElt ] → GrpAb
  • AbelianGroup(G): Grp → GrpAb, Hom
  • AbelianQuotient(G): Grp → GrpAb, Hom
  • DirectSum(A, B): GrpAb, GrpAb → GrpAb
  • PCGroup(A): GrpAb → GrpPC, Hom(Grp)
  • PermutationGroup(A): GrpAb → GrpPerm, Hom(Grp)
  • FPGroup(A): GrpAb → GrpFP, Hom(Grp)
  • CommutatorSubgroup(G): GrpAb → GrpAb
  • DerivedSubgroup(G): GrpAb → GrpAb
  • DerivedGroup(G): GrpAb → GrpAb
  • CommutatorSubgroup(H, K): GrpAb, GrpAb → GrpAb
  • CommutatorSubgroup(G, H, K): GrpAb, GrpAb, GrpAb → GrpAb
  • Centralizer(G, a): GrpAb, GrpAbElt → GrpAb
  • Centraliser(G, a): GrpAb, GrpAbElt → GrpAb
  • Core(G, H): GrpAb, GrpAb → GrpAb
  • Centre(G): GrpAb → GrpAb
  • Center(G): GrpAb → GrpAb
  • FittingGroup(G): GrpAb → GrpAb
  • FittingSubgroup(G): GrpAb → GrpAb
  • Hypercentre(G): GrpAb → GrpAb
  • Hypercenter(G): GrpAb → GrpAb
• Operations on Elements
  • Order of an Element
    • Order(x): GrpAbElt → RngIntElt
    • Example: Discrete Log
    • Order(g: parameters): GrpAbGenElt → RngIntElt
    • Order(g, l, u: parameters): GrpAbGenElt, RngIntElt, RngIntElt → RngIntElt
    • Order(g, l, u, n, m: parameters): GrpAbGenElt, RngIntElt, RngIntElt, RngIntElt, RngIntElt → RngIntElt
  • Discrete Logarithm
    • Log(g, d: parameters): GrpAbGenElt, GrpAbGenElt → RngIntElt
    • Example: Discrete Log
  • Equality and Comparison
    • u eq v: GrpAbElt, GrpAbElt → BoolElt
    • u ne v: GrpAbElt, GrpAbElt → BoolElt
    • IsIdentity(u): GrpAbElt → BoolElt
    • IsId(u): GrpAbElt → BoolElt
• Invariants of an Abelian Group
  • ElementaryAbelianQuotient(G, p): GrpAb, RngIntElt → GrpAb, Map
  • FreeAbelianQuotient(G): GrpAb → GrpAb, Map
  • PrimaryAbelianInvariants(A): GrpAb → [ RngIntElt ]
  • AbelianInvariants(A): GrpAb → [ RngIntElt ]
  • PrimaryAbelianBasis(A): GrpAb → [ GrpAbElt ], [ RngIntElt ]
  • AbelianBasis(A): GrpAb → [ GrpAbElt ], [ RngIntElt ]
  • TorsionFreeRank(A): GrpAb → RngIntElt
  • TorsionInvariants(A): GrpAb → [ RngIntElt ]
  • pPrimaryInvariants(A, p): GrpAb, RngIntElt → [ RngIntElt ]
• Canonical Decomposition
  • TorsionFreeSubgroup(A): GrpAb → GrpAb
  • TorsionSubgroup(A): GrpAb → GrpAb
  • pPrimaryComponent(A, p): GrpAb, RngIntElt → GrpAb
• Set-Theoretic Operations
  • Functions Relating to Group Order
    • Order(G): GrpAb → RngIntElt
    • # G: GrpAb → RngIntElt
    • FactoredOrder(G): GrpAb → [<RngIntElt, RngIntElt>]
    • Exponent(G): GrpAb → RngIntElt
    • IsFinite(G): GrpAb → BoolElt
    • IsInfinite(G): GrpAb → BoolElt
  • Membership and Equality
    • g in G: GrpAbElt, GrpAb → BoolElt
    • g notin G: GrpAbElt, GrpAb → BoolElt
    • S subset G: { GrpAbElt}, GrpAb → BoolElt
    • S notsubset G: { GrpAbElt}, GrpAb → BoolElt
    • H subset G: GrpAb, GrpAb → BoolElt
    • IsSubgroup(H,G): GrpAb, GrpAb → BoolElt
    • H notsubset G: GrpAb, GrpAb → BoolElt
    • G eq H: GrpAb, GrpAb → BoolElt
    • G ne H: GrpAb, GrpAb → BoolElt
  • Set Operations
    • NumberingMap(G): GrpAb → Map
    • RandomProcess(G): GrpAb → Process
    • Random(P): Process → GrpAbElt
    • Random(G): GrpAb → GrpAbElt
    • Random(G, n): GrpAb, RngIntElt → GrpAbElt
    • Rep(G): GrpAb → GrpAbElt
    • Representative(G): GrpAb → GrpAbElt
• Coset Spaces
  • Transversal(G, H): GrpAb, GrpAb → {\@ GrpAbElt \@}, Map
  • RightTransversal(G, H): GrpAb, GrpAb → {\@ GrpAbElt \@}, Map
  • Coercions Between Groups and Subgroups
    • G ! g: GrpAb, GrpAbElt → GrpAbElt
    • H ! g: GrpAb, GrpAbElt → GrpAbElt
    • K ! g: GrpAb, GrpAbElt → GrpAbElt
    • Morphism(H, G): GrpAb, GrpAb → ModMatRngElt
• Subgroup Constructions
  • H meet K: GrpAb, GrpAb → GrpAb
  • H meet:= K: GrpAb, GrpAb → GrpAb
  • H + K: GrpAb, GrpAb → GrpAb
  • n * G: RngIntElt, GrpAb → GrpAb, Map
  • FrattiniSubgroup(G): GrpAb → GrpAb
  • SylowSubgroup(G, p : parameters): GrpAb, RngIntElt → GrpAb
  • Sylow(G, p : parameters): GrpAb, RngIntElt → GrpAb
  • Example: p Sylow Computation
• Subgroup Chains
  • ChiefSeries(G): GrpAb → [GrpAb]
  • CompositionSeries(G): GrpAb → [GrpAb]
  • Agemo(G, i): GrpAb, RngIntElt → GrpAb
  • Omega(G, i): GrpAb, RngIntElt → GrpAb
  • ElementaryAbelianSeries(G): GrpAb → [ GrpAb ]
  • DerivedSeries(G): GrpAb → [ GrpAb ]
  • LowerCentralSeries(G): GrpAb → [ GrpAb ]
  • UpperCentralSeries(G): GrpAb → [ GrpAb ]
  • SubnormalSeries(G, H): GrpAb, GrpAb → [ GrpAb ]
• General Group Properties
  • IsCyclic(G): GrpAb → BoolElt
  • IsElementaryAbelian(G): GrpAb → BoolElt
  • IsFree(G): GrpAb → BoolElt
  • IsMixed(G): GrpAb → BoolElt
  • IspGroup(G): GrpAb → BoolElt
  • DerivedLength(G): GrpAb → RngIntElt
  • Properties of Subgroups
    • IsMaximal(G, H): GrpAb, GrpAb → BoolElt
    • Index(G, H): GrpAb, GrpAb → RngIntElt
    • FactoredIndex(G, H): GrpAb, GrpAb → [<RngIntElt, RngIntElt>]
    • IsPure(G, H): GrpAb, GrpAb → BoolElt
    • IsNeat(G, H): GrpAb, GrpAb → BoolElt
  • Enumeration of Subgroups
    • MaximalSubgroups(G): GrpAb → [GrpAb]
    • Subgroups(G:parameters): GrpAb → [Rec]
    • NumberOfSubgroupsAbelianPGroup(A): SeqEnum → SeqEnum
    • HasComplement(G, U): GrpAb, GrpAb → BoolElt, GrpAb
    • Example: Subgroups
• Representation Theory
  • CharacterTable(G): GrpAb → TabChtr
• The Hom Functor
  • Hom(G, H): GrpPC, GrpPC → GrpAb, Map
  • Hom(G, H): GrpAb, GrpAb → GrpAb, Map
  • HomGenerators(G, H): GrpAb, GrpAb → GrpAb, Map
  • AllHomomorphisms(G, H): GrpAb, GrpAb → [Map]
  • AllHomomorphisms(G, H): GrpPC, GrpPC → [Map]
  • Homomorphisms(G, H): GrpAb, GrpAb → [Map]
  • Example: Relations
• Automorphism Groups
  • AutomorphismGroup(G): GrpAb → GrpAuto
  • IsIsomorphic(G, H): GrpAb, GrpAb → BoolElt, Hom(Grp)
• Cohomology
  • Dual(G): GrpAb → GrpAb, Map
  • H2_G_QmodZ(G): GrpAb → GrpAb, Map
  • Res_H2_G_QmodZ(U, H2): GrpAb, GrpAb → GrpAb, Map
• Homomorphisms
  • hom< A -> B | L>: Grp, Grp, List → Map
  • Homomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] → Map
  • iso< A -> B | L>: Grp, Grp, List → Map
  • Isomorphism(A, B, X, Y): Grp, Grp, [ GrpElt ], [ GrpElt ] → Map
  • Example: Homomorphisms