The Categories of Finite Groups
Construction of an Element
elt< G | L > : Grp, List(Elt) -> GrpElt
G ! Q : Grp, [ Elt ] -> GrpElt
Identity(G) : Grp -> GrpElt
Coercion
G ! g : Grp, GrpElt -> GrpElt
Homomorphisms
hom< G -> H | L > : Grp, Grp -> Map
hom< G -> H | x : -> e(x) > : Grp, Grp -> Map
IdentityHomomorphism(G) : Grp -> Map
Example Grp_Homomorphisms (H57E1)
Example Grp_Homomorphisms-2 (H57E2)
Arithmetic with Elements
g * h : GrpElt, GrpElt -> GrpElt
g ^ n : GrpElt, RngIntElt -> GrpElt
g / h : GrpElt, GrpElt -> GrpElt
g ^ h : GrpElt, GrpElt -> GrpElt
(g, h) : GrpElt, GrpElt -> GrpElt
(g1, ..., gr) : GrpElt, ..., GrpElt -> GrpElt
g eq h : GrpElt, GrpElt -> BoolElt
g ne h : GrpElt, GrpElt -> BoolElt
IsId(g) : GrpElt -> BoolElt
Order(g) : GrpElt -> RngIntElt
Example Grp_Arithmetic (H57E3)
Construction of a General Group
The General Group Constructors
PermutationGroup< X | L > : Set, List -> GrpPerm, Hom
Group< X | R > : List(Identifiers), List(GrpFPRel) -> GrpFP, Hom(Grp)
Example Grp_GroupConstructors (H57E4)
Example Grp_PolycyclicGroup (H57E5)
Construction of Subgroups
sub<G | L> : Grp, List -> Grp
ncl<G | L> : Grp, List -> Grp
Example Grp_Subgroup (H57E6)
Construction of Quotient Groups
quo<G | L> : Grp, List -> Grp, Map
G / N : Grp, Grp -> Grp
Example Grp_Quotient (H57E7)
Standard Groups and Extensions
Construction of a Standard Group
AbelianGroup(C, Q) : Cat, [ RngIntElt ] -> GrpFin
AlternatingGroup(C, n) : Cat, RngIntElt -> GrpFin
CyclicGroup(C, n) : Cat, RngIntElt -> GrpFin
DihedralGroup(C, n) : Cat, RngIntElt -> GrpFin
DicyclicGroup(n) : RngIntElt -> GrpFP
SymmetricGroup(C, n) : Cat, RngIntElt -> GrpFin
ExtraSpecialGroup(C, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpFin
Example Grp_StandardGroups (H57E8)
Construction of Extensions
DirectProduct(G, H) : Grp, Grp -> Grp
DirectProduct(Q) : [ Grp ] -> Grp
SemidirectProduct(K, H, f: parameters) : Grp, Grp, Map -> Grp, Map, Map
Example Grp_Extensions (H57E9)
Transfer Functions Between Group Categories
pQuotient(F, p, c: parameters) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map
CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, Grp
CosetImage(G, H) : Grp, Grp -> GrpPerm
CosetKernel(G, H) : Grp, Grp -> Grp
GPCGroup(G) : Grp -> GrpGPC, Hom(Grp)
PCGroup(G) : Grp -> GrpPC, Hom(Grp)
FPGroup(G: parameters) : GrpPerm :-> GrpFP, Hom(Grp)
Example Grp_CosetAction (H57E10)
Example Grp_CosetAction-2 (H57E11)
Example Grp_FPGroup (H57E12)
Accessing Group Information
G . i : Grp, RngIntElt -> GrpElt
Generators(G) : Grp -> { GrpFinElt }
NumberOfGenerators(G) : Grp -> RngIntElt
Generic(G) : Grp -> Grp
Parent(g) : GrpElt -> Grp
Example Grp_Generators (H57E13)
Orbit(G, M, x) : Grp, Any, Any -> Any
OrbitClosure(G, M, S) : Grp, Any, Any -> Any
Operations on the Set of Elements
Order and Index Functions
Order(G) : GrpFin -> RngIntElt
FactoredOrder(G) : GrpFin -> [ <RngIntElt, RngIntElt> ]
Index(G, H) : GrpFin, GrpFin -> RngIntElt
FactoredIndex(G, H) : GrpFin, GrpFin -> [ <RngIntElt, RngIntElt> ]
Example Grp_Order (H57E14)
Membership and Equality
g in G : GrpFinElt, GrpFin -> BoolElt
g notin G : GrpFinElt, GrpFin -> BoolElt
S subset G : { GrpFinElt }, GrpFin -> BoolElt
S notsubset G : { GrpFinElt }, GrpFin -> BoolElt
H subset G : GrpFin, GrpFin -> BoolElt
H notsubset G : GrpFin, GrpFin -> BoolElt
H eq G : GrpFin, GrpFin -> BoolElt
H ne G : GrpFin, GrpFin -> BoolElt
Set Operations
NumberingMap(G) : GrpFin -> Map
Representative(G) : GrpFin -> GrpFinElt
Example Grp_SetOperations (H57E15)
Random Elements
Random(G: parameters) : GrpFin -> GrpFinElt
Example Grp_RandomOperations (H57E16)
RandomProcess(G) : GrpFin -> Process
Random(P) : Process -> GrpFinElt
InitialiseProspector(G:parameters): GrpMat ->
Prospector(G, f:parameters): Grp, UserProgram -> BoolElt, GrpElt, GrpSLPElt
Example Grp_RandomProspector (H57E17)
Action on a Coset Space
CosetTable(G, H) : GrpFin, GrpFin -> Map
[Future release] CosetTable(G, f) : GrpFin, Hom(GrpFin) -> Hom(GrpFin)
Transversal(G, H) : Grp, Grp -> {@ GrpElt @}, Map
CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, Grp
CosetImage(G, H) : Grp, Grp -> GrpPerm
CosetKernel(G, H) : Grp, Grp -> Grp
Standard Subgroup Constructions
H ^ g : GrpFin, GrpFinElt -> GrpFin
H meet K : GrpFin, GrpFin -> GrpFin
CommutatorSubgroup(G, H, K) : GrpFin, GrpFin, GrpFin -> GrpFin
Centralizer(G, g) : GrpFin, GrpFinElt -> GrpFin
Centralizer(G, H) : GrpFin, GrpFin -> GrpFin
Core(G, H) : GrpFin, GrpFin -> GrpFin
H ^ G : GrpFin, GrpFin -> GrpFin
Normalizer(G, H) : GrpFin, GrpFin -> GrpFin
pCore(G, p) : GrpFin, RngIntElt -> GrpFin
SylowSubgroup(G, p) : GrpFin, RngIntElt -> GrpFin
Abstract Group Predicates
IsAbelian(G) : GrpFin -> BoolElt
IsCyclic(G) : GrpFin -> BoolElt
IsElementaryAbelian(G) : GrpFin -> BoolElt
IsCentral(G, H) : GrpFin -> BoolElt
IsConjugate(G, g, h) : GrpFin, GrpFinElt, GrpFinElt -> BoolElt, GrpFinElt
IsConjugate(G, H, K) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt
IsExtraSpecial(G) : GrpFin -> BoolElt
IsMaximal(G, H) : GrpFin, GrpFin -> BoolElt
IsNilpotent(G) : GrpFin -> BoolElt
IsNormal(G, H) : GrpFin, GrpFin -> BoolElt
IsPerfect(G) : GrpFin -> BoolElt
IsSelfNormalizing(G, H) : GrpFin, GrpFin -> BoolElt
IsSimple(G) : GrpFin -> BoolElt
IsSoluble(G) : GrpFin -> BoolElt
IsSpecial(G) : GrpFin -> BoolElt
IsSubnormal(G, H) : GrpFin, GrpFin -> BoolElt
IsTrivial(G) : Grp -> BoolElt
Characteristic Subgroups and Normal Structure
Characteristic Subgroups and Subgroup Series
Centre(G) : GrpFin -> GrpFin
Hypercentre(G) : GrpFin -> GrpFin
DerivedLength(G) : GrpFin -> RngIntElt
DerivedSeries(G) : GrpFin -> [ GrpFin ]
DerivedSubgroup(G) : GrpFin -> GrpFin
FittingSubgroup(G) : GrpFin -> GrpFin
FrattiniSubgroup(G) : GrpFin -> GrpFin
JenningsSeries(G) : GrpFin -> [ GrpFin ]
LowerCentralSeries(G) : GrpFin -> [ GrpFin ]
NilpotencyClass(G) : GrpFin -> RngIntElt
H ^ G : GrpFin -> GrpFin
NormalLattice(G) : GrpFin -> NormalLattice
NormalSubgroups(G) : GrpFin -> [ Rec ]
pCentralSeries(G, p) : GrpFin, RngIntElt -> [ GrpFin ]
Radical(G) : GrpFin -> GrpFin
SolubleResidual(G) : GrpFin -> GrpFin
SubnormalSeries(G, H) : GrpFin, GrpFin -> [ GrpFin ]
UpperCentralSeries(G) : GrpFin -> [ GrpFin ]
The Abstract Structure of a Group
CompositionFactors(G) : : GrpFin -> [ <RngIntElt, RngIntElt, RngIntElt> ]
AbelianInvariants(G) : GrpFin -> [ RngIntElt ]
AbelianBasis(G) : GrpFin -> [ GrpFinElt ], [ RngIntElt ]
Conjugacy Classes of Elements
Class(H, x) : GrpFin, GrpFinElt -> { GrpFinElt }
ClassMap(G: parameters) : GrpFin -> Map
ConjugacyClasses(G: parameters) : GrpFin -> [ <RngIntElt, RngIntElt, GrpFinElt> ]
ClassRepresentative(G, x) : GrpFin, GrpFinElt -> GrpFinElt
IsConjugate(G, g, h) : GrpFin, GrpFinElt, GrpFinElt -> BoolElt, GrpFinElt
IsConjugate(G, H, K) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt
Exponent(G) : GrpFin -> RngIntElt
NumberOfClasses(G) : GrpFin -> RngIntElt
PowerMap(G) : GrpFin -> Map
Example Grp_Classes (H57E18)
Conjugacy Classes of Subgroups
Conjugacy Classes of Subgroups
SubgroupClasses(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
ElementaryAbelianSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
AbelianSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
CyclicSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
NilpotentSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
SolubleSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
NonsolvableSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
PerfectSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
SimpleSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
RegularSubgroups(G: parameters) : GrpFin -> [ rec< Grp, RngIntElt, RngIntElt, GrpFP> ]
SetVerbose("SubgroupLattice", i) : MonStgElt, RngIntElt ->
Class(G, H) : GrpFin, GrpFin -> { GrpFin }
Example Grp_Subgroups (H57E19)
Creating the Poset of Subgroup Classes
SubgroupLattice(G) : GrpFin -> SubGrpLat
Example Grp_CreateSubgroupPoset (H57E20)
Operations on Subgroup Class Posets
# L : SubGrpLat -> RngIntElt
L ! i: SubGrpLat, RngIntElt -> SubGrpLatElt
L ! H: SubGrpLat, GrpFin -> SubGrpLatElt
Bottom(L): SubGrpLat -> SubGrpLatElt
Top(L): SubGrpLat -> SubGrpLatElt
Random(L): SubGrpLat -> SubGrpLatElt
Example Grp_LatticeOperations (H57E21)
Operations on Poset Elements
IntegerRing() ! e : SubGrpLatElt -> RngIntElt
e eq f : SubGrpLatElt, SubGrpLatElt -> SubGrpLatElt
e ge f : SubGrpLatElt, SubGrpLatElt -> BoolElt
e ge f : SubGrpLatElt, SubGrpLatElt -> BoolElt
e le f : SubGrpLatElt, SubGrpLatElt -> BoolElt
e lt f : SubGrpLatElt, SubGrpLatElt -> BoolElt
Class Information from a Conjugacy Class Poset
Group(e) : SubGrpLatElt -> GrpFin
Centraliser(e, f) : SubGrpLatElt, SubGrpLatElt -> SubGrpLatElt
Normaliser(e, f) : SubGrpLatElt, SubGrpLatElt -> SubGrpLatElt
Length(e) : SubGrpLatElt -> RngIntElt
Order(e) : SubGrpLatElt -> RngIntElt
MaximalSubgroups(e) : SubGrpLatElt -> { SubGrpLatElt }
MinimalOvergroups(e) : SubGrpLatElt -> { SubGrpLatElt }
NumberOfInclusions(e, f) : SubGrpLatElt, SubGrpLatElt -> RngIntElt
Cohomology
pMultiplicator(G, p) : GrpFin, RngIntElt -> [ RngIntElt ]
pCover(G, F, p) : GrpPerm, GrpFP, RngIntElt -> GrpFinFP
CohomologicalDimension(G, M, i) : GrpFin, ModRng, RngIntElt -> RngIntElt
ExtensionProcess(G, M, F) : GrpPerm, ModRng, GrpFP -> GrpFPExtProc
Extension(P, Q) : Process -> GrpFinFP
SplitExtension(G, M, F) : GrpPerm, ModRng, GrpFP -> GrpFP
Characters and Representations
Character Theory
CharacterDegrees(G) : GrpPC -> [ Tup ]
CharacterTable(G) : GrpFin -> TabChtr
PermutationCharacter(G) : GrpPerm -> AlgChtrElt
PermutationCharacter(G, H) : GrpFin, GrpFin -> AlgChtrElt
Representation Theory
GModule(G, S) : GrpFin, AlgMat -> ModGrpFin
GModule(G, A, B) : GrpFin, GrpFin, GrpFin -> ModGrpFin, Map
PermutationModule(G, H, R) : GrpFin, GrpFin, Rng -> ModGrpFin
PermutationModule(G, R) : GrpPerm, Rng -> ModGrpFin
Example Grp_Modules (H57E22)
Example Grp_Modules-2 (H57E23)
Bibliography
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Mon Dec 17 14:40:36 EST 2012