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ABELIAN GROUPS

 
Acknowledgements
 
Introduction
 
Construction of a Finitely Presented Abelian Group and its Elements
      The Free Abelian Group
      Relations
      Specification of a Presentation
      Accessing the Defining Generators and Relations
 
Construction of a Generic Abelian Group
      Specification of a Generic Abelian Group
      Accessing Generators
      Computing Abelian Group Structure
 
Elements
      Construction of Elements
      Representation of an Element
      Arithmetic with Elements
 
Construction of Subgroups and Quotient Groups
      Construction of Subgroups
      Construction of Quotient Groups
 
Standard Constructions and Conversions
 
Operations on Elements
      Order of an Element
      Discrete Logarithm
      Equality and Comparison
 
Invariants of an Abelian Group
 
Canonical Decomposition
 
Set-Theoretic Operations
      Functions Relating to Group Order
      Membership and Equality
      Set Operations
 
Coset Spaces
      Coercions Between Groups and Subgroups
 
Subgroup Constructions
 
Subgroup Chains
 
General Group Properties
      Properties of Subgroups
      Enumeration of Subgroups
 
Representation Theory
 
The Hom Functor
 
Automorphism Groups
 
Cohomology
 
Homomorphisms
 
Bibliography







DETAILS

 
Introduction

 
Construction of a Finitely Presented Abelian Group and its Elements

      The Free Abelian Group
            FreeAbelianGroup(n) : RngIntElt -> GrpAb
            Example GrpAb_FreeAbelianGroup (H69E1)

      Relations
            w1 = w2 : GrpAbElt, GrpAbElt -> Rel
            r[1] : GrpAbRel, RngIntElt -> GrpAbElt
            r[2] : GrpAbRel, RngIntElt -> GrpAbElt
            Parent(r) : RelElt -> GrpAb
            Example GrpAb_Relations (H69E2)

      Specification of a Presentation
            AbelianGroup< X | R > : List(Var), List(GrpAbRel) -> GrpAb, Hom(GrpAb)
            Example GrpAb_AbelianGroup (H69E3)
            AbelianGroup([n1,...,nr]): [ RngIntElt ] -> GrpAb
            Example GrpAb_AbelianGroup2 (H69E4)

      Accessing the Defining Generators and Relations
            A . i : GrpAb, RngIntElt -> GrpAbElt
            Generators(A) : GrpAb -> { GrpAbElt }
            NumberOfGenerators(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 GrpAb_Creation (H69E5)

      Accessing Generators
            Universe(A) : GrpAbGen ->
            A . i : GrpAbGen, RngIntElt -> GrpAbGenElt
            Generators(A) : GrpAbGen -> [ GrpAbGenElt ]
            UserGenerators(A) : GrpAbGen -> [ GrpAbGenElt ]
            NumberOfGenerators(A) : GrpAbGen -> RngIntElt

      Computing Abelian Group Structure
            AbelianGroup(A: parameters) : GrpAbGen -> GrpAb, Map
            Example GrpAb_GroupComputation (H69E6)

 
Elements

      Construction of Elements
            A ! [a1, ... ,an] : GrpAb, [RngIntElt] -> GrpAbElt
            A ! e : GrpAbGen, Elt -> GrpAbGenElt
            A ! g : GrpAbGen, GrpAbGenElt -> GrpAbGenElt
            A ! n : GrpAb, RngIntElt -> GrpAbElt
            Random(A) : GrpAbGen -> GrpAbGenElt
            Identity(A) : GrpAb -> GrpAbElt

      Representation of an Element
            Representation(g) : GrpAbGenElt -> [RngIntElt]
            UserRepresentation(g) : GrpAbGenElt -> [RngIntElt]
            Representation(S, g) : SeqEnum, GrpAbGenElt -> [RngIntElt], RngIntElt
            Example GrpAb_ElementCreationAndRep (H69E7)

      Arithmetic with Elements
            u + v : GrpAbElt, GrpAbElt -> GrpAbElt
            - u : GrpAbElt -> GrpAbElt
            u - v : GrpAbElt, GrpAbElt -> GrpAbElt
            m * u : RngIntElt, GrpAbElt-> GrpAbElt

 
Construction of Subgroups and Quotient Groups

      Construction of Subgroups
            sub<A | L> : GrpAb, List -> GrpAb, Map
            Example GrpAb_SubgroupCreation (H69E8)
            sub<A | L: parameters> : GrpAbGen, List -> GrpAbGen
            Example GrpAb_GenericSubgroupCreation (H69E9)

      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(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
      CommutatorSubgroup(H, K) : GrpAb, GrpAb -> GrpAb
      Centralizer(G, a) : GrpAb, GrpAbElt -> GrpAb
      Core(G, H) : GrpAb, GrpAb -> GrpAb
      Centre(G) : GrpAb -> GrpAb

 
Operations on Elements

      Order of an Element
            Order(x) : GrpAbElt -> RngIntElt
            Example GrpAb_DiscreteLog (H69E10)
            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 GrpAb_DiscreteLog (H69E11)

      Equality and Comparison
            u eq v : GrpAbElt, GrpAbElt -> BoolElt
            u ne v : GrpAbElt, GrpAbElt -> BoolElt
            IsIdentity(u) : GrpAbElt -> BoolElt

 
Invariants of an Abelian Group
      ElementaryAbelianQuotient(G, p) : GrpAb, RngIntElt -> GrpAb, Map
      FreeAbelianQuotient(G) : GrpAb -> GrpAb, Map
      Invariants(A) : GrpAb -> [ RngIntElt ]
      TorsionFreeRank(A) : GrpAb -> RngIntElt
      TorsionInvariants(A) : GrpAb -> [ RngIntElt ]
      PrimaryInvariants(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
            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
            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
            Rep(G) : GrpAb -> GrpAbElt

 
Coset Spaces
      Transversal(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
      Example GrpAb_pSylowComputation (H69E12)

 
Subgroup Chains
      CompositionSeries(G) : GrpAb -> [GrpAb]
      Agemo(G, i) : GrpAb, RngIntElt -> GrpAb
      Omega(G, i) : GrpAb, RngIntElt -> GrpAb

 
General Group Properties
      IsCyclic(G) : GrpAb -> BoolElt
      IsElementaryAbelian(G) : GrpAb -> BoolElt
      IsFree(G) : GrpAb -> BoolElt
      IsMixed(G) : GrpAb -> BoolElt
      IspGroup(G) : GrpAb -> BoolElt

      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 GrpAb_Subgroups (H69E13)

 
Representation Theory
      CharacterTable(G) : GrpAb -> TabChtr

 
The Hom Functor
      Hom(G, H) : GrpPC, GrpPC -> GrpAb, Map
      HomGenerators(G, H) : GrpAb, GrpAb -> GrpAb, Map
      AllHomomorphisms(G, H) : GrpAb, GrpAb -> [Map]
      Example GrpAb_Relations (H69E14)

 
Automorphism Groups
      AutomorphismGroup(G) : GrpAb -> GrpAuto

 
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 GrpAb_Homomorphisms (H69E15)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012