[____]
GROUPS
Acknowledgements
Introduction
The Categories of Finite Groups
Construction of Elements
Construction of an Element
Coercion
Homomorphisms
Arithmetic with Elements
Construction of a General Group
The General Group Constructors
Construction of Subgroups
Construction of Quotient Groups
Standard Groups and Extensions
Construction of a Standard Group
Construction of Extensions
Transfer Functions Between Group Categories
Basic Operations
Accessing Group Information
Names of Finite Groups
Operations on the Set of Elements
Order and Index Functions
Membership and Equality
Set Operations
Random Elements
Action on a Coset Space
Standard Subgroup Constructions
Abstract Group Predicates
Characteristic Subgroups and Normal Structure
Characteristic Subgroups and Subgroup Series
The Abstract Structure of a Group
Conjugacy Classes of Elements
Conjugacy Classes of Subgroups
Conjugacy Classes of Subgroups
The Poset of Subgroup Classes
Creating the Poset of Subgroup Classes
Operations on Subgroup Class Posets
Operations on Poset Elements
Class Information from a Conjugacy Class Poset
All Subgroups and Intermediate Subgroups
Cohomology
Characters and Representations
Character Theory
Representation Theory
Databases of Groups
Bibliography
Introduction
The Categories of Finite Groups
Construction of Elements
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 (H64E1)
Example Grp_Homomorphisms-2 (H64E2)
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 (H64E3)
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 (H64E4)
Example Grp_PolycyclicGroup (H64E5)
Construction of Subgroups
sub<G | L> : Grp, List -> Grp
ncl<G | L> : Grp, List -> Grp
Example Grp_Subgroup (H64E6)
Construction of Quotient Groups
quo<G | L> : Grp, List -> Grp, Map
G / N : Grp, Grp -> Grp
Example Grp_Quotient (H64E7)
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 (H64E8)
Construction of Extensions
DirectProduct(G, H) : Grp, Grp -> Grp
DirectProduct(Q) : [ Grp ] -> Grp
SemidirectProduct(K, H, f: parameters) : Grp, Grp, Map -> Grp, Map, Map, Map
Example Grp_semidirect (H64E9)
AffineSplitExtension(M: parameters) : ModGrp -> Grp, Map, Map, Map
Example Grp_affine-split (H64E10)
Example Grp_Extensions (H64E11)
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
MinimalDegreePermutationRepresentation(G: parameters) : Grp -> Hom(Grp), GrpPerm
Example Grp_MinimalDegreePermutationRepresentation (H64E12)
GPCGroup(G) : Grp -> GrpGPC, Hom(Grp)
PCGroup(G) : Grp -> GrpPC, Hom(Grp)
FPGroup(G: parameters) : GrpPerm :-> GrpFP, Hom(Grp)
Example Grp_CosetAction (H64E13)
Example Grp_CosetAction-2 (H64E14)
Example Grp_FPGroup (H64E15)
Basic Operations
Accessing Group Information
G . i : Grp, RngIntElt -> GrpElt
Generators(G) : Grp -> { GrpFinElt }
NumberOfGenerators(G) : Grp -> RngIntElt
SmallestGeneratingSet(G: parameters) : Grp -> SetIndx
Generic(G) : Grp -> Grp
Parent(g) : GrpElt -> Grp
Example Grp_Generators (H64E16)
Orbit(G, M, x) : Grp, Any, Any -> Any
OrbitClosure(G, M, S) : Grp, Any, Any -> Any
Names of Finite Groups
GroupName(G) : Grp -> MonStgElt
Example Grp_grp-groupname (H64E17)
Group(s) : MonStgElt -> Grp
Example Grp_grp-group (H64E18)
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 (H64E19)
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 (H64E20)
Random Elements
Random(G: parameters) : GrpFin -> GrpFinElt
Example Grp_RandomOperations (H64E21)
RandomProcess(G) : GrpFin -> Process
Random(P) : Process -> GrpFinElt
Random(P) : Process -> GrpFinElt
InitialiseProspector(G:parameters): GrpMat ->
Prospector(G, f:parameters): Grp, UserProgram -> BoolElt, GrpElt, GrpSLPElt
Example Grp_RandomProspector (H64E22)
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, GrpFin -> BoolElt
IsConjugate(G, g, h) : GrpFin, GrpFinElt, GrpFinElt -> BoolElt, GrpFinElt
IsConjugate(G, H, K) : GrpFin, GrpFin, GrpFin -> BoolElt, GrpFinElt
IsExtraSpecial(G) : GrpFin -> BoolElt
IsHyperelementary(G) : Grp -> BoolElt,Grp,Grp
Example Grp_grp-ishyperelementary (H64E23)
IsMaximal(G, H) : GrpFin, GrpFin -> BoolElt
IsNilpotent(G) : GrpFin -> BoolElt
IsNormal(G, H) : GrpFin, GrpFin -> BoolElt
IsPerfect(G) : GrpFin -> BoolElt
IsQGroup(G) : Grp -> BoolElt
Example Grp_grp-isqgroup (H64E24)
IsSelfNormalizing(G, H) : GrpFin, GrpFin -> BoolElt
IsSimple(G) : GrpFin -> BoolElt
IsSoluble(G) : GrpFin -> BoolElt
IsSpecial(G) : GrpFin -> BoolElt
IsAbstractFrobeniusGroup(G) : GrpFin -> BoolElt, Grp, Grp
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> ]
PrimaryAbelianInvariants(G) : GrpFin -> [ RngIntElt ]
PrimaryAbelianBasis(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> ]
ClassesData(G: parameters) : GrpFin -> [ <RngIntElt, RngIntElt> ]
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 (H64E25)
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 (H64E26)
The Poset of Subgroup Classes
Creating the Poset of Subgroup Classes
SubgroupLattice(G) : GrpFin -> SubGrpLat
Example Grp_CreateSubgroupPoset (H64E27)
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 (H64E28)
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
All Subgroups and Intermediate Subgroups
AllSubgroups(G) : GrpFin -> [ SeqEnum ]
MinimalOvergroups(G,H) : GrpFin, GrpFin -> [ SeqEnum ]
IntermediateSubgroups(G,H: parameters) : GrpFin, GrpFin -> [ SeqEnum ]
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
BurnsideCokernel(G) : Grp -> GrpAb, UserProgram, SeqEnum[AlgChtrElt]
Example Grp_grp-burnsidecokernel (H64E29)
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 (H64E30)
Example Grp_Modules-2 (H64E31)
Databases of Groups
Bibliography
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