MAGMA Computational Algebra System

ABELIAN GROUPS

DETAILS

 
Introduction

 
Construction of a Finitely Presented Abelian Group and its Elements

      The Free Abelian Group
            FreeAbelianGroup(n) : RngIntElt -> GrpAb
            Example GrpAb_FreeAbelianGroup (H60E1)

      Relations
            w₁ = w₂ : GrpAbElt, GrpAbElt -> Rel
            LHS(r) : Rel -> GrpAbElt
            RHS(r) : Rel -> GrpAbElt
            Parent(r) : RelElt -> GrpAb
            Example GrpAb_Relations (H60E2)

      Specification of a Presentation
            AbelianGroup< X | R > : List(Var), List(GrpAbRel) -> GrpAb, Hom(GrpAb)
            Example GrpAb_AbelianGroup (H60E3)
            AbelianGroup([n₁,...,n_r]): [ RngIntElt ] -> GrpAb
            Example GrpAb_AbelianGroup2 (H60E4)

      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 (H60E5)

      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 (H60E6)

 
Elements

      Construction of Elements
            A ! [a₁, ... ,a_n] : 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 (H60E7)

      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 (H60E8)
            sub<A | L: parameters> : GrpAbGen, List -> GrpAbGen
            Example GrpAb_GenericSubgroupCreation (H60E9)

      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)

 
Operations on Elements

      Order of an Element
            Order(x) : GrpAbElt -> RngIntElt
            Example GrpAb_DiscreteLog (H60E10)
            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 (H60E11)

      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 (H60E12)

 
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 (H60E13)

 
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 (H60E14)

 
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

 
Bibliography

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