Polycyclic Groups and Polycyclic Presentations
Specification of Elements
G ! Q : GrpGPC, [RngIntElt] -> GrpGPCElt
Identity(G) : GrpGPC -> GrpGPCElt
Access Functions for Elements
ElementToSequence(x) : GrpGPCElt -> [RngIntElt]
LeadingTerm(x) : GrpGPCElt -> GrpGPCElt
LeadingGenerator(x) : GrpGPCElt -> GrpGPCElt
LeadingExponent(x) : GrpGPCElt -> RngIntElt
Depth(x) : GrpGPCElt -> RngIntElt
Arithmetic Operations on Elements
g * h : GrpGPCElt, GrpGPCElt -> GrpGPCElt
g *:= h : GrpGPCElt, GrpGPCElt ->
g ^ n: GrpGPCElt, RngIntElt -> GrpGPCElt
g ^:= n: GrpGPCElt, RngIntElt ->
g / h : GrpGPCElt, GrpGPCElt -> GrpGPCElt
g /:= h : GrpGPCElt, GrpGPCElt ->
g ^ h : GrpGPCElt, GrpGPCElt -> GrpGPCElt
g ^:= h : GrpGPCElt, GrpGPCElt ->
(g1, ..., gn) : List(GrpGPCElt) -> GrpGPCElt
Operators for Elements
Order(x) : GrpGPCElt -> RngIntElt
Parent(x) : GrpGPCElt -> GrpGPC
Comparison Operators for Elements
g eq h : GrpGPCElt, GrpGPCElt -> BoolElt
g ne h : GrpGPCElt, GrpGPCElt -> BoolElt
IsIdentity(g) : GrpGPCElt -> BoolElt
Specification of a Polycyclic Presentation
quo< GrpGPC : F | R : parameters > : GrpFP, List(GrpFPRel) -> GrpGPC, Map
PolycyclicGroup< x1, ..., xn | R : parameters > : List(Identifiers), List(GrpFPRel) -> GrpGPC, Map
Example GrpGPC_Constructor (H72E1)
Example GrpGPC_PolycyclicGroup (H72E2)
Properties of a Polycyclic Presentation
IsConsistent(G) : GrpGPC -> BoolElt
IsIdenticalPresentation(G, H) : GrpGPC, GrpGPC -> BoolElt
PresentationIsSmall(G) : GrpGPC -> BoolElt
Subgroups, Quotient Groups, Homomorphisms and Extensions
Construction of Subgroups
sub<G | L> : GrpGPC, List -> GrpGPC, Map
ncl<G | L> : GrpGPC, List -> GrpGPC, Map
Coercions Between Groups and Subgroups
G ! g : GrpGPC, GrpGPCElt -> GrpGPCElt
H ! g : GrpGPC, GrpGPCElt -> GrpGPCElt
K ! g : GrpGPC, GrpGPCElt -> GrpGPCElt
InclusionMap(G, H) : GrpGPC, GrpGPC -> Map
Example GrpGPC_Subgroup (H72E3)
Construction of Quotient Groups
quo<G | L> : GrpGPC, List -> GrpGPC, Map
G / N : GrpGPC, GrpGPC -> GrpGPC
Construction of Homomorphisms
hom< P -> G | S : parameters> : Struct , Struct -> Map
Construction of Extensions
DirectProduct(G, H) : GrpGPC, GrpGPC -> GrpGPC, [Map], [Map]
Construction of Standard Groups
AbelianGroup(GrpGPC, Q) : Cat, [RngIntElt] -> GrpGPC
CyclicGroup(GrpGPC, n) : Cat, RngIntElt -> GrpGPC
DihedralGroup(GrpGPC, n) : Cat, RngIntElt -> GrpGPC
ElementaryAbelianGroup(GrpGPC, p, n) : Cat, RngIntElt, RngIntElt -> GrpGPC
ExtraSpecialGroup(GrpGPC, p, n : parameters) : Cat, RngIntElt, RngIntElt -> GrpGPC
FreeAbelianGroup(GrpGPC, n) : Cat, RngIntElt -> GrpGPC
FreeNilpotentGroup(r, e) : RngIntElt, RngIntElt -> GrpGPC
Example GrpGPC_Homomorphism (H72E4)
Example GrpGPC_Symmetric2 (H72E5)
Conversion between Categories
AbelianGroup(G) : GrpGPC -> GrpAb, Map
FPGroup(G) : GrpGPC -> GrpFP, Map
PCGroup(G) : GrpGPC -> GrpPC, Map
GPCGroup(G) : GrpPerm -> GrpGPC, Map
Example GrpGPC_SubgroupsQuotientsTransfer (H72E6)
Access Functions for Groups
G . i : GrpGPC, RngIntElt -> GrpGPCElt
Generators(G) : GrpGPC -> {@ GrpGPCElt @}
Generators(H, G) : GrpGPC, GrpGPC -> {@ GrpGPCElt @}
NumberOfGenerators(G) : GrpGPC -> RngIntElt
PCExponents(G) : GrpGPC -> [RngIntElt]
HirschNumber(G) : GrpGPC -> RngIntElt
Set-Theoretic Operations in a Group
Functions Relating to Group Order
FactoredIndex(G, H) : GrpGPC, GrpGPC -> [<RngIntElt, RngIntElt>]
FactoredOrder(G) : GrpGPC -> [<RngIntElt, RngIntElt>]
Index(G, H) : GrpGPC, GrpGPC -> RngIntElt
Order(G) : GrpGPC -> RngIntElt
Membership and Equality
g in G : GrpGPCElt, GrpGPC -> BoolElt
g notin G : GrpGPCElt, GrpGPC -> BoolElt
S subset G : { GrpGPCElt } , GrpGPC -> BoolElt
S notsubset G : { GrpGPCElt } , GrpGPC -> BoolElt
H subset G : GrpGPC, GrpGPC -> BoolElt
H notsubset G : GrpGPC, GrpGPC -> BoolElt
G eq H : GrpGPC, GrpGPC -> BoolElt
G ne H : GrpGPC, GrpGPC -> BoolElt
Set Operations
Representative(G) : GrpGPC -> GrpGPCElt
RandomProcess(G) : GrpGPC -> Process
Random(P) : Process -> GrpGPCElt
Random(G) : GrpGPC -> GrpGPCElt
Coset Spaces
CosetTable(G, H) : GrpGPC, GrpGPC -> Map
Transversal(G, H) : GrpGPC, GrpGPC -> {@ GrpGPCElt @}, Map
Example GrpGPC_CosetTable (H72E7)
CosetAction(G, H) : GrpGPC, GrpGPC -> Map, GrpPerm, GrpGPC
CosetImage(G, H) : GrpGPC, GrpGPC -> GrpPerm
CosetKernel(G, H) : GrpGPC, GrpGPC -> GrpGPC
Example GrpGPC_CosetAction (H72E8)
General Subgroup Constructions
H ^ g : GrpGPC, GrpGPCElt -> GrpGPC
H ^ G : GrpGPC, GrpGPC -> GrpGPC
CommutatorSubgroup(G, H, K) : GrpGPC, GrpGPC, GrpGPC -> GrpGPC
Subgroup Constructions Requiring a Nil-po-tent Covering Group
H meet K : GrpGPC, GrpGPC -> GrpGPC
H meet:= K : GrpGPC, GrpGPC -> GrpGPC
Centraliser(G, g) : GrpGPC, GrpGPCElt -> GrpGPC
Centraliser(G, H) : GrpGPC, GrpGPC -> GrpGPC
Core(G, H) : GrpGPC, GrpGPC -> GrpGPC
Normaliser(G, H) : GrpGPC, GrpGPC -> GrpGPC
General Group Properties
IsAbelian(G) : GrpGPC -> BoolElt
IsCyclic(G) : GrpGPC -> BoolElt
IsElementaryAbelian(G) : GrpGPC -> BoolElt
IsFinite(G) : GrpGPC -> BoolElt
IsNilpotent(G) : GrpGPC -> BoolElt
IsPerfect(G) : GrpGPC -> BoolElt
IsSimple(G) : GrpGPC -> BoolElt
IsSoluble(G) : GrpGPC -> BoolElt
General Properties of Subgroups
IsCentral(G, H) : GrpGPC, GrpGPC -> BoolElt
IsNormal(G, H) : GrpGPC, GrpGPC -> BoolElt
Properties of Subgroups Requiring a Nil-po-tent Covering Group
IsConjugate(G, H, K) : GrpGPC, GrpGPC, GrpGPC -> BoolElt, GrpGPCElt
IsSelfNormalising(G, H) : GrpGPC, GrpGPC -> BoolElt
Example GrpGPC_SubgroupStructure (H72E9)
Example GrpGPC_SubgroupStructure2 (H72E10)
Normal Structure and Characteristic Subgroups
Characteristic Subgroups and Subgroup Series
Centre(G) : GrpGPC -> GrpGPC
DerivedLength(G) : GrpGPC -> RngIntElt
DerivedSeries(G) : GrpGPC -> [GrpGPC]
DerivedSubgroup(G) : GrpGPC -> GrpGPC
EFASeries(G) : GrpGPC -> [GrpGPC]
FittingLength(G) : GrpGPC -> RngIntElt
FittingSeries(G) : GrpGPC -> [GrpGPC]
FittingSubgroup(G) : GrpGPC -> GrpGPC
HasComputableLCS(G) : GrpGPC -> BoolElt
LowerCentralSeries(G) : GrpGPC -> [GrpGPC]
NilpotencyClass(G) : GrpGPC -> RngIntElt
NilpotentPresentation(G) : GrpGPC -> GrpGPC, Map
SemisimpleEFASeries(G) : GrpGPC -> [GrpGPC]
UpperCentralSeries(G) : GrpGPC -> [GrpGPC]
Example GrpGPC_NormalStructure (H72E11)
The Abelian Quotient Structure of a Group
AbelianQuotient(G) : GrpGPC -> GrpAb, Map
AbelianQuotientInvariants(G) : GrpGPC -> [ RngIntElt ]
ElementaryAbelianQuotient(G, p) : GrpGPC, RngIntElt -> GrpAb, Map
FreeAbelianQuotient(G) : GrpGPC -> GrpAb, Map
Conjugacy
IsConjugate(G, g, h) : GrpGPC, GrpGPCElt, GrpGPCElt -> BoolElt, GrpGPCElt
IsConjugate(G, H, K) : GrpGPC, GrpGPC, GrpGPC -> BoolElt, GrpGPCElt
Example GrpGPC_Conjugacy (H72E12)
Representation Theory
EFAModuleMaps(G) : GrpGPC -> [ModGrp]
EFAModules(G) : GrpGPC -> [ModGrp]
GModule(G, A, p) : GrpGPC, GrpGPC, RngIntElt -> ModGrp, Map
GModule(G, A, B, p) : GrpGPC, GrpGPC, GrpGPC, RngIntElt -> ModGrp, Map
GModulePrimes(G, A) : GrpGPC, GrpGPC -> SetMulti
GModulePrimes(G, A, B) : GrpGPC, GrpGPC, GrpGPC -> SetMulti
SemisimpleEFAModuleMaps(G) : GrpGPC -> [ModGrp]
SemisimpleEFAModules(G) : GrpGPC -> [ModGrp]
Example GrpGPC_RepresentationTheory (H72E13)
Example GrpGPC_gmoduleprimes (H72E14)
Example GrpGPC_FittingSubgroup (H72E15)
Example GrpGPC_ModuleMaps (H72E16)
Power Groups
Parent(G) : GrpGPC -> PowStr
PowerGroup(G) : GrpPC -> PowerGroup
Bibliography
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Version: V2.19 of
Mon Dec 17 14:40:36 EST 2012