[____]
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
Introduction
Construction of a Finitely Presented Abelian Group and its Elements
The Free Abelian Group
FreeAbelianGroup(n) : RngIntElt -> GrpAb
Example GrpAb_FreeAbelianGroup (H77E1)
Relations
w1 = w2 : GrpAbElt, GrpAbElt -> Rel
r[1] : GrpAbRel, RngIntElt -> GrpAbElt
r[2] : GrpAbRel, RngIntElt -> GrpAbElt
Parent(r) : RelElt -> GrpAb
Example GrpAb_Relations (H77E2)
Specification of a Presentation
AbelianGroup< X | R > : List(Var), List(GrpAbRel) -> GrpAb, Hom(GrpAb)
Example GrpAb_AbelianGroup (H77E3)
AbelianGroup([n1,...,nr]): [ RngIntElt ] -> GrpAb
Example GrpAb_AbelianGroup2 (H77E4)
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 (H77E5)
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 (H77E6)
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 (H77E7)
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 (H77E8)
sub<A | L: parameters> : GrpAbGen, List -> GrpAbGen
Example GrpAb_GenericSubgroupCreation (H77E9)
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
FittingGroup(G) : GrpAb -> GrpAb
Hypercentre(G) : GrpAb -> GrpAb
Operations on Elements
Order of an Element
Order(x) : GrpAbElt -> RngIntElt
Example GrpAb_DiscreteLog (H77E10)
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 (H77E11)
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
PrimaryAbelianInvariants(A) : GrpAb -> [ RngIntElt ]
PrimaryAbelianBasis(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
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 (H77E12)
Subgroup Chains
ChiefSeries(G) : GrpAb -> [GrpAb]
Agemo(G, i) : GrpAb, RngIntElt -> GrpAb
Omega(G, i) : GrpAb, RngIntElt -> GrpAb
ElementaryAbelianSeries(G) : GrpAb -> [ GrpAb ]
DerivedSeries(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 GrpAb_Subgroups (H77E13)
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 (H77E14)
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 GrpAb_Homomorphisms (H77E15)
Bibliography
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