POLYCYCLIC GROUPS
Acknowledgements
Introduction
Polycyclic Groups and Polycyclic Presentations
Introduction
Specification of Elements
Access Functions for Elements
Arithmetic Operations on Elements
Operators for Elements
Comparison Operators for Elements
Specification of a Polycyclic Presentation
Properties of a Polycyclic Presentation
Subgroups, Quotient Groups, Homomorphisms and Extensions
Construction of Subgroups
Coercions Between Groups and Subgroups
Construction of Quotient Groups
Homomorphisms
General remarks
Construction of Homomorphisms
Construction of Extensions
Construction of Standard Groups
Conversion between Categories
Access Functions for Groups
Set-Theoretic Operations in a Group
Functions Relating to Group Order
Membership and Equality
Set Operations
Coset Spaces
The Subgroup Structure
General Subgroup Constructions
Subgroup Constructions Requiring a Nil-po-tent Covering Group
General Group Properties
General Properties of Subgroups
Properties of Subgroups Requiring a Nil-po-tent Covering Group
Normal Structure and Characteristic Subgroups
Characteristic Subgroups and Subgroup Series
The Abelian Quotient Structure of a Group
Conjugacy
Representation Theory
Power Groups
Bibliography
Introduction
Polycyclic Groups and Polycyclic Presentations
Introduction
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
IsFinite(x) : GrpGPCElt -> BoolElt
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 (H81E1)
Example GrpGPC_PolycyclicGroup (H81E2)
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 (H81E3)
Construction of Quotient Groups
quo<G | L> : GrpGPC, List -> GrpGPC, Map
G / N : GrpGPC, GrpGPC -> GrpGPC
Homomorphisms
General remarks
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 (H81E4)
Example GrpGPC_Symmetric2 (H81E5)
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 (H81E6)
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 (H81E7)
CosetAction(G, H) : GrpGPC, GrpGPC -> Map, GrpPerm, GrpGPC
CosetImage(G, H) : GrpGPC, GrpGPC -> GrpPerm
CosetKernel(G, H) : GrpGPC, GrpGPC -> GrpGPC
Example GrpGPC_CosetAction (H81E8)
The Subgroup Structure
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 (H81E9)
Example GrpGPC_SubgroupStructure2 (H81E10)
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 (H81E11)
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 (H81E12)
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 (H81E13)
Example GrpGPC_gmoduleprimes (H81E14)
Example GrpGPC_FittingSubgroup (H81E15)
Example GrpGPC_ModuleMaps (H81E16)
Power Groups
Parent(G) : GrpGPC -> PowStr
PowerGroup(G) : GrpPC -> PowerGroup
Bibliography
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