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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







DETAILS

 
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
            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

      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 (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)

 
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 (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 Wed Apr 24 15:09:57 EST 2013