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)
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
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
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