Construction Functions
CyclicGroup(GrpPC, n) : Cat, RngIntElt -> GrpPC
AbelianGroup(GrpPC, Q) : Cat, [RngIntElt] -> GrpPC
DihedralGroup(GrpPC, n) : Cat, RngIntElt -> GrpPC
ExtraSpecialGroup(GrpPC, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpPC
Example GrpPC_Standard (H63E1)
Definition by Presentation
PolycyclicGroup< x1, ..., xn | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpPC, Map
quo< GrpPC : F | R : parameters > : GrpFP, List(GrpFPRel) -> GrpPC, Map
Example GrpPC_PolycyclicGroup (H63E2)
Possibly Inconsistent Presentations
IsConsistent(G) : GrpPC -> BoolElt
Example GrpPC_IsConsistent (H63E3)
Infrastructure
G . i : GrpPC, RngIntElt -> GrpPCElt
Generators(G) : GrpPC -> SetEnum
NumberOfGenerators(G) : GrpPC -> RngIntElt
PCGenerators(G) : GrpPC -> SetIndx
NumberOfPCGenerators(G) : GrpPC -> RngIntElt
PCPrimes(G) : GrpPC -> [RngIntElt]
Numerical Invariants
Order(G) : GrpPC -> RngIntElt
FactoredOrder(G) : GrpPC -> [<RngIntElt, RngIntElt>]
Exponent(G) : GrpPC -> RngIntElt
Predicates
IsAbelian(G) : GrpPC -> BoolElt
IsCyclic(G) : GrpPC -> BoolElt
IsElementaryAbelian(G) : GrpPC -> BoolElt
IsNilpotent(G) : GrpPC -> BoolElt
IsPerfect(G) : GrpPC -> BoolElt
IsSimple(G) : GrpPC -> BoolElt
IsSoluble(G) : GrpPC -> BoolElt
IsTrivial(G) : GrpPC -> BoolElt
IsSpecial(G) : GrpPC -> BoolElt
IsExtraSpecial(G) : GrpPC -> BoolElt
Example GrpPC_group-props (H63E4)
Homomorphisms
hom< G -> H | L > : GrpPC, GrpPC, List -> Map
IsHomomorphism(G, H, L) : GrpPC, GrpPC, SeqEnum -> BoolElt, Map
IdentityHomomorphism(G) : GrpPC -> Map
Kernel(f) : Map -> GrpPC
Homomorphisms(G, H) : GrpPC, GrpPC -> SeqEnum
Example GrpPC_pc_hom (H63E5)
New Groups from Existing
DirectProduct(G, H) : GrpPC, GrpPC -> GrpPC, [Map], [Map]
DirectProduct(Q) : [GrpPC] -> GrpPC, [ Map ], [ Map ]
Extension(G, H, f) : GrpPC, GrpPC, [Map] -> GrpPC
Extension(M, H) : ModGrp, GrpPC -> GrpPC
Extension(G, H, f, t) : GrpPC, GrpPC, [Map], [GrpPCElt] -> GrpPC
Extension(M, H, t) : ModGrp, GrpPC, [ModGrpElt] -> GrpPC
IsExtension(G, H, f) : GrpPC, GrpPC, [Map] -> BoolElt, GrpPC
WreathProduct(G, H) : GrpPC, GrpPC -> GrpPC
WreathProduct(G, H, f) : GrpPC, GrpPC, Map -> GrpPC
Example GrpPC_extension (H63E6)
Example GrpPC_cossey_hawkes (H63E7)
Definition of Elements
G ! Q : GrpPC, [RngIntElt] -> GrpPCElt
ElementToSequence(x) : GrpPCElt -> [RngIntElt]
Identity(G) : GrpPC -> GrpPCElt
Example GrpPC_elt-definition (H63E8)
Arithmetic Operations on Elements
g * h : GrpPCElt, GrpPCElt -> GrpPCElt
g *:= h : GrpPCElt, GrpPCElt -> GrpPCElt
g ^ n: GrpPCElt, RngIntElt -> GrpPCElt
g ^:= n: GrpPCElt, RngIntElt -> GrpPCElt
g / h : GrpPCElt, GrpPCElt -> GrpPCElt
g /:= h : GrpPCElt, GrpPCElt -> GrpPCElt
g ^ h : GrpPCElt, GrpPCElt -> GrpPCElt
g ^:= h : GrpPCElt, GrpPCElt -> GrpPCElt
(g1, ..., gn) : List(GrpPCElt) -> GrpPCElt
Properties of Elements
Order(x) : GrpPCElt -> RngIntElt
Parent(x) : GrpPCElt -> GrpPC
Predicates for Elements
g eq h : GrpPCElt, GrpPCElt -> BoolElt
g ne h : GrpPCElt, GrpPCElt -> BoolElt
IsIdentity(g) : GrpPCElt -> BoolElt
IsConjugate(G, g, h) : GrpPC, GrpPCElt, GrpPCElt -> BoolElt, GrpPCElt
Example GrpPC_elt_predicates (H63E9)
Set Operations
NumberingMap(G) : GrpPC -> Map
Random(G) : GrpPC -> GrpPCElt
RandomProcess(G) : GrpPC -> Process
Random(P) : Process -> GrpPCElt
Representative(G) : GrpPC -> GrpPCElt
Example GrpPC_set_ops (H63E10)
Example GrpPC_Set (H63E11)
Conjugacy
Class(H, g) : GrpPC, GrpPCElt -> { GrpPCElt }
ConjugacyClasses(G) : GrpPC -> [ <RngIntElt, RngIntElt, GrpPCElt> ]
ClassMap(G) : GrpPC -> Map
ClassRepresentative(G, x) : GrpPC, GrpPCElt -> GrpPCElt
IsConjugate(G, g, h) : GrpPC, GrpPCElt, GrpPCElt -> BoolElt, GrpPCElt
NumberOfClasses(G) : GrpPC -> RngIntElt
PowerMap(G) : GrpPC -> Map
Example GrpPC_class_map (H63E12)
Definition of Subgroups by Generators
sub<G | L> : GrpPC, List -> GrpPC, Map
ncl<G | L> : GrpPC, List -> GrpPC, Map
Example GrpPC_sub_creation (H63E13)
Membership and Coercion
g in G : GrpPCElt, GrpPC -> BoolElt
g notin G : GrpPCElt, GrpPC -> BoolElt
G ! g : GrpPC, GrpPCElt -> GrpPCElt
H ! g : GrpPC, GrpPCElt -> GrpPCElt
K ! g : GrpPC, GrpPCElt -> GrpPCElt
Example GrpPC_coercion (H63E14)
Inclusion and Equality
S subset G : { GrpPCElt } , GrpPC -> BoolElt
S notsubset G : { GrpPCElt } , GrpPC -> BoolElt
H subset G : GrpPC, GrpPC -> BoolElt
H notsubset G : GrpPC, GrpPC -> BoolElt
G eq H : GrpPC, GrpPC -> BoolElt
G ne H : GrpPC, GrpPC -> BoolElt
InclusionMap(G, H) : GrpPC, GrpPC -> Map
Standard Subgroup Constructions
H ^ g : GrpPC, GrpPCElt -> GrpPC
H meet K : GrpPC, GrpPC -> GrpPC
H meet:= K : GrpPC, GrpPC -> GrpPC
CommutatorSubgroup(G, H, K) : GrpPC, GrpPC, GrpPC -> GrpPC
Centralizer(G, g) : GrpPC, GrpPCElt -> GrpPC
Centralizer(G, H) : GrpPC, GrpPC -> GrpPC
Core(G, H) : GrpPC, GrpPC -> GrpPC
H ^ G : GrpPC, GrpPC -> GrpPC
Normalizer(G, H) : GrpPC, GrpPC -> GrpPC
Example GrpPC_subgroup-constructions (H63E15)
Properties of Subgroups
Index(G, H) : GrpPC, GrpPC -> RngIntElt
FactoredIndex(G, H) : GrpPC, GrpPC -> [<RngIntElt, RngIntElt>]
Predicates for Subgroups
IsCentral(G, H) : GrpPC, GrpPC -> BoolElt
IsConjugate(G, H, K) : GrpPC, GrpPC, GrpPC -> BoolElt, GrpPCElt
IsMaximal(G, H) : GrpPC, GrpPC -> BoolElt
IsNormal(G, H) : GrpPC, GrpPC -> BoolElt
IsSelfNormalizing(G, H) : GrpPC, GrpPC -> BoolElt
IsSubnormal(G, H) : GrpPC, GrpPC -> BoolElt
Example GrpPC_sub-predicates (H63E16)
Hall π-Subgroups and Sylow Systems
ComplementBasis(G) : GrpPC -> [GrpPC]
HallSubgroup(G, S) : GrpPC, { RngIntElt } -> GrpPC
pCore(G, S) : GrpPC, { RngIntElt } -> GrpPC
SylowBasis(G) : GrpPC -> [GrpPC]
SylowSubgroup(G, p) : GrpPC, RngIntElt -> GrpPC
SystemNormalizer(G) : GrpPC -> GrpPC
Example GrpPC_Hall (H63E17)
Conjugacy Classes of Subgroups
SubgroupClasses(G) : GrpPC -> SeqEnum
AbelianSubgroups(G) : GrpPC -> SeqEnum
MaximalSubgroups(G) : GrpPC -> [GrpPC]
SubgroupLattice(G) : GrpPC -> SubGrpLat
BurnsideMatrix(G) : GrpPC -> AlgMatElt
DisplayBurnsideMatrix(G) : GrpPC ->
Example GrpPC_SubgroupClasses (H63E18)
Construction of Quotient Groups
quo<G | L> : GrpPC, List -> GrpPC, Map
G / N : GrpPC, GrpPC -> GrpPC
Example GrpPC_pc_quotient (H63E19)
Abelian and p-Quotients
AbelianQuotient(G) : GrpPC -> GrpAb, Map
AbelianQuotientInvariants(G) : GrpPC -> SeqEnum
ElementaryAbelianQuotient(G, p) : GrpPC, RngIntElt -> GrpAb, Map
pQuotient(G, p, c : parameters ) : GrpPC, RngIntElt, RngIntElt -> GrpPC, Map
Normal Subgroups and Subgroup Series
Characteristic Subgroups
Centre(G) : GrpPC -> GrpPC
CommutatorSubgroup(G) : GrpPC -> GrpPC
FittingSubgroup(G) : GrpPC -> GrpPC
FrattiniSubgroup(G) : GrpPC -> GrpPC
Hypercentre(G) : GrpPC -> GrpPC
MinimalNormalSubgroups(G) : GrpPC -> [GrpPC]
pCore(G, S) : GrpPC, { RngIntElt } -> GrpPC
Socle(G) : GrpPC -> GrpPC
Subgroup Series
AbelianBasis(G) : GrpPC -> [ GrpPCElt ], [ RngIntElt ]
AbelianInvariants(G) : GrpPC -> [RngIntElt]
ChiefSeries(G) : GrpPC -> [GrpPC]
CompositionSeries(G) : GrpPC -> [GrpPC]
CompositionFactors(G) : GrpPC -> SeqEnum
CompositionSeries(G, i) : GrpPC, RngIntElt -> [GrpPC]
DerivedSeries(G) : GrpPC -> [GrpPC]
DerivedLength(G) : GrpPC -> RngIntElt
ElementaryAbelianSeries(G) : GrpPC -> [GrpPC]
ElementaryAbelianSeriesCanonical(G) : GrpPC -> [GrpPC]
LowerCentralSeries(G) : GrpPC -> [GrpPC]
NilpotencyClass(G) : GrpPC -> RngIntElt
pCentralSeries(G, p) : GrpPC, RngIntElt -> [GrpPC]
SubnormalSeries(G, H) : GrpPC, GrpPC -> [GrpPC]
UpperCentralSeries(G) : GrpPC -> [GrpPC]
Example GrpPC_EAS (H63E20)
Series for p-groups
Agemo(G, i) : GrpPC, RngIntElt -> GrpPC
Omega(G, i) : GrpPC, RngIntElt -> GrpPC
JenningsSeries(G) : GrpPC -> [GrpPC]
pClass(G) : GrpPC -> RngIntElt
pRanks(G) : GrpPC-> [ RngIntElt ]
Normal Subgroups and Complements
NormalSubgroups(G) : GrpPC -> SeqEnum
NormalLattice(G) : GrpPC -> SubGrpLat
MinimalNormalSubgroup(G) : GrpPC -> GrpPC
MinimalNormalSubgroup(G, N) : GrpPC -> GrpPC
Complements(G, N) : GrpPC, GrpPC -> SeqEnum
NormalComplements(G, N) : GrpPC, GrpPC -> SeqEnum
NormalComplements(G, H, N) : GrpPC, GrpPC -> SeqEnum
Example GrpPC_NormalComplements (H63E21)
Coset Tables and Transversals
Transversal(G, H) : GrpPC, GrpPC -> {@ GrpPCElt @}, Map
CosetTable(G, H) : GrpPC, GrpPC -> Map
Transversal(G, H, K) : GrpPC, GrpPC, GrpPC -> {@ GrpPCElt @}, Map
ShortCosets(p, H, G) : GrpPCElt, GrpPC, GrpPC -> [GrpPCElt]
Action on a Coset Space
CosetAction(G, H) : Grp, Grp -> Hom(Grp), GrpPerm, GrpPC
CosetImage(G, H) : Grp, Grp -> GrpPerm
CosetKernel(G, H) : Grp, Grp -> Grp
Lifting Algorithm
AutomorphismGroup(G): GrpPC -> GrpAuto
HasAttribute(A, "GenWeights") : GrpAuto, MonStgElt -> BoolElt, [ RngIntElt ]
HasAttribute(A, "WeightSubgroupOrders") : GrpAuto, MonStgElt -> BoolElt, [ RngIntElt ]
Example GrpPC_AutomorphismGroup (H63E22)
Lifting from the Automorphism Group of a Sylow p-subgroup
AutomorphismGroupSolubleGroup(G: parameters): GrpPC -> GrpAuto
IsIsomorphicSolubleGroup(G, H: parameters) : GrpPC, GrpPC -> BoolElt, Map
Example GrpPC_AutomorphismGroupSolubleGroup (H63E23)
p-group
AutomorphismGroup(G: parameters): GrpPC -> GrpAuto
Example GrpPC_pAutomorphismGroup (H63E24)
OrderAutomorphismGroupAbelianPGroup (A) : SeqEnum -> RngIntElt
Example GrpPC_subgroupsabelianpgroups (H63E25)
Isomorphism and Standard Presentations
StandardPresentation(G): GrpPC -> GrpPC, Map
IsIdenticalPresentation(G, H) : GrpPC, GrpPC -> BoolElt
IsIsomorphic(G, H) : GrpPC, GrpPC -> BoolElt, Map
Example GrpPC_StandardPresentation (H63E26)
Generating p-groups
GeneratepGroups (p, d, c : parameters) : RngIntElt, RngIntElt,RngIntElt -> [GrpPC], RngIntElt
Descendants(G : parameters) : GrpPC -> [GrpPC], RngIntElt
Example GrpPC_Generating_p_groups (H63E27)
Example GrpPC_GeneratepGroups (H63E28)
Example GrpPC_IsGood (H63E29)
ClassTwo(p, d : parameters) : RngIntElt, RngIntElt -> SeqEnum
Example GrpPC_ClassTwo (H63E30)
Representation Theory
CharacterDegrees(G) : GrpPC -> [ Tup ]
CharacterDegrees(G) : GrpFin -> [ Tup ]
CharacterDegreesPGroup(G) : GrpFin -> [ RngIntElt ]
CharacterTable(G: parameters) : GrpPC -> TabChtr
CharacterTableConlon(G) : GrpPC -> [ AlgChtrElt ]
GModule(G, M) : GrpPC, AlgMat -> ModAlg
GModule(G, A) : GrpPC, GrpPC -> ModAlg, Map
GModule(G, A, B) : GrpPC, GrpPC, GrpPC -> ModAlg, Map
AbsolutelyIrreducibleRepresentationsSchur(G, k: parameters) : GrpPC, Rng -> List[Map]
IrreducibleRepresentationsSchur(G, k: parameters) : GrpPC, Rng -> List[Map]
Example GrpPC_Reps (H63E31)
Central Extensions
ExtGenerators(G, U) : GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]
HomGenerators(G, U) : GrpPC, GrpPC -> [<AlgMatElt, RngIntElt>]
ElementSequence(G) : GrpPC -> SeqEnum
RepresentativeCocycles(G, U, Ext, Hom) : GrpPC, GrpPC, [AlgMatElt], [AlgMatElt]-> [AlgMatElt]
CentralExtension(G, U, A) : GrpPC, GrpPC, AlgMatElt -> GrpPC
CentralExtensions(G, U, Q) : GrpPC, GrpPC, [AlgMatElt] -> [GrpPC]
CentralExtensionProcess(G, U) : GrpPC, GrpPC -> Proc
NextExtension(~P) : Rec -> GrpPC
IsEmpty(P) : Rec -> BoolElt
Example GrpPC_CentralExtension (H63E32)
Transfer Between Group Categories
Transfer to GrpPC
PCGroup(G) : GrpPerm -> GrpPC, Map
pQuotient( F, p, c : parameters ) : GrpFP, RngIntElt, RngIntElt -> GrpPC, Map
SolubleQuotient(G) : Grp -> GrpPC, Map
Example GrpPC_pcgroup (H63E33)
Transfer from GrpPC
AbelianGroup(G) : GrpPC -> GrpAb, Map
FPGroup(G) : GrpPC -> GrpFP, Map
GPCGroup(G) : GrpPC -> GrpGPC, Map
Example GrpPC_pc-to-perm (H63E34)
Structure Operations
ConditionedGroup(G) : GrpPC -> GrpPC
IsConditioned(G) : GrpPC -> BoolElt
Element Operations
LeadingTerm(x) : GrpPCElt -> GrpPCElt
LeadingGenerator(x) : GrpPCElt -> GrpPCElt
LeadingExponent(x) : GrpPCElt -> RngIntElt
Depth(x) : GrpPCElt -> RngIntElt
PCClass(x) : GrpPCElt -> RngIntElt
Special Presentations
SpecialPresentation(G) : GrpPC -> GrpPC
SpecialWeights(G) : GrpPC -> [ <RngIntElt, RngIntElt, RngIntElt> ]
NilpotentLength(G) : GrpPC -> RngIntElt
NilpotentBoundary(G,i) : GrpPC, RngIntElt -> RngIntElt
MinorLength(G,i) : GrpPC, RngIntElt -> RngIntElt
MinorBoundary(G,i,j) : GrpPC, RngIntElt, RngIntElt -> RngIntElt
LayerLength(G,i,j) : GrpPC, RngIntElt, RngIntElt -> RngIntElt
LayerBoundary(G,i,j,k) : GrpPC, RngIntElt, RngIntElt, RngIntElt -> RngIntElt
Example GrpPC_SpecialPresentation (H63E35)
CompactPresentation
CompactPresentation(G) : GrpPC -> [RngIntElt]
PCGroup(Q : parameters ) : [RngIntElt] -> GrpPC
Example GrpPC_CompactPresentation (H63E36)
PowerGroup
Example GrpPC_PowerGroupTwo (H63E37)
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012