Construction of a Free Abelian Group and its Elements
Structure Constructors
FreeAbelianGroup(n) : RngIntElt -> GrpAb
Example GrpAb_FreeAbelianGroup (H29E1)
Construction of an Element
A ! [a_1, ... ,a_n] : GrpAb, [RngIntElt] -> GrpAbElt
A ! n : GrpAb, RngIntElt -> GrpAbElt
Identity(A) : GrpAb -> GrpAbElt
Deconstruction of an Element
ElementToSequence(x) : GrpAbElt -> [RngIntElt]
Addition and Subtraction
u + v : GrpAbElt, GrpAbElt -> GrpAbElt
- u : GrpAbElt -> GrpAbElt
u - v : GrpAbElt, GrpAbElt -> GrpAbElt
m * u : RngIntElt, GrpAbElt-> GrpAbElt
Relations
w_1 = w_2 : GrpAbElt, GrpAbElt -> Rel
LHS(r) : Rel -> GrpAbElt
RHS(r) : Rel -> GrpAbElt
Parent(r) : GrpAbRel -> GrpAb
Example GrpAb_Relations (H29E2)
Construction of Subgroups and Quotient Groups
Construction of Subgroups
sub<A | L> : GrpAb, List -> GrpAb, Map
Construction of Quotient Groups
quo<F | R> : GrpAb, List -> GrpAb, Hom(GrpAb)
A / B : GrpAb, GrpAb -> GrpAb
Specification of a Presentation
AbelianGroup< X | R > : List(Var), List(GrpAbRel) -> GrpAb, Hom(GrpAb)
Example GrpAb_AbelianGroup (H29E3)
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 ]
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)
Order of an Element
Order(x) : GrpAbElt -> RngIntElt
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>]
Index(G, H) : GrpAb, GrpAb -> RngIntElt
FactoredIndex(G, H) : GrpAb, GrpAb -> [<RngIntElt, RngIntElt>]
Exponent(G) : GrpAb -> RngIntElt
IsFinite(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
The Subgroup Structure
H ^ g : GrpAb, GrpAbElt -> GrpAb
H ^ G : GrpAb, GrpAb -> GrpAb
H meet K : GrpAb, GrpAb -> GrpAb
H meet:= K : GrpAb, GrpAb -> GrpAb
CommutatorSubgroup(G, H, K) : GrpAb, GrpAb, GrpAb -> GrpAb
Centralizer(G, g) : GrpAb, GrpAbElt -> GrpAb
Centralizer(G, H) : GrpAb, GrpAb -> GrpAb
Core(G, H) : GrpAb, GrpAb -> GrpAb
NormalClosure(G, H) : GrpAb, GrpAb -> GrpAb
Normalizer(G, H) : GrpAb, GrpAb -> GrpAb
SylowSubgroup(G, p) : GrpAb, RngIntElt -> GrpAb
pCore(G, p) : GrpAb, RngIntElt -> GrpAb
General Group Properties
IsAbelian(G) : GrpAb -> BoolElt
IsCyclic(G) : GrpAb -> BoolElt
IsElementaryAbelian(G) : GrpAb -> BoolElt
IsNilpotent(G) : GrpAb -> BoolElt
IsPerfect(G) : GrpAb -> BoolElt
IsSimple(G) : GrpAb -> BoolElt
IsSoluble(G) : GrpAb -> BoolElt
General Properties of Subgroups
IsCentral(G, H) : GrpAb, GrpAb -> BoolElt
IsConjugate(G, H, K) : GrpAb, GrpAb, GrpAb -> BoolElt, GrpAbElt
IsMaximal(G, H) : GrpAb, GrpAb -> BoolElt
IsNormal(G, H) : GrpAb, GrpAb -> BoolElt
IsSubnormal(G, H) : GrpAb, GrpAb -> BoolElt
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
Normal Structure and Characteristic Subgroups
Characteristic Subgroups and Subgroup Series
Agemo(G, i) : GrpAb, RngIntElt -> GrpAb
Centre(G) : GrpAb -> GrpAb
ChiefSeries(G) : GrpAb -> [GrpAb]
DerivedLength(G) : GrpAb -> RngIntElt
DerivedSeries(G) : GrpAb -> [GrpAb]
DerivedSubgroup(G) : GrpAb -> GrpAb
ElementaryAbelianSeries(G) : GrpAb -> [GrpAb]
FittingSubgroup(G) : GrpAb -> GrpAb
FrattiniSubgroup(G) : GrpAb -> GrpAb
Hypercentre(G) : GrpAb -> GrpAb
NilpotencyClass(G) : GrpAb -> RngIntElt
Omega(G, i) : GrpAb, RngIntElt -> GrpAb
SubnormalSeries(G, H) : GrpAb, GrpAb -> [GrpAb]
UpperCentralSeries(G) : GrpAb -> [GrpAb]
Subgroup Structure
MaximalSubgroups(G) : GrpAb -> [GrpAb]
Subgroups(G:parameters) : GrpAb -> [Rec]
Example GrpAb_Subgroups (H29E4)
Conjugacy
Class(H, g) : GrpAb, GrpAbElt -> { GrpAbElt }
ConjugacyClasses(G) : GrpAb -> [ <RngIntElt, RngIntElt, GrpAbElt> ]
ClassMap(G) : GrpAb -> Map
ClassMatrix(G, i) : GrpAb, RngIntElt -> AlgMatElt
ClassRepresentative(G, x) : GrpAb, GrpAbElt -> GrpAbElt
IsConjugate(G, g, h) : GrpAb, GrpAbElt, GrpAbElt -> BoolElt, GrpAbElt
IsConjugate(G, H, K) : GrpAb, GrpAb, GrpAb -> BoolElt, GrpAbElt
NumberOfClasses(G) : GrpAb -> RngIntElt
PowerMap(G) : GrpAb -> Map
Representation Theory
CharacterTable(G) : GrpAb -> TabChtr
Computation of Hom
Hom(G, H) : GrpAb, GrpAb -> GrpAb, Map
HomGenerators(G, H) : GrpAb, GrpAb -> GrpAb, Map
Homomorphisms(G, H) : GrpAb, GrpAb -> GrpAb, Map
Example GrpAb_Relations (H29E5)
Cohomology
Dual(G) : GrpAb -> GrpAb, Map
H2_G_QmodZ(G) : GrpAb -> GrpAb, Map