Polycyclic Groups

• Introduction
• Polycyclic Groups and Polycyclic Presentations
  • Introduction
  • Specification of Elements
    • G ! Q: GrpGPC, [RngIntElt] → GrpGPCElt
    • Identity(G): GrpGPC → GrpGPCElt
    • Id(G): GrpGPC → GrpGPCElt
    • G ! 1: GrpGPC, RngIntElt → GrpGPCElt
  • Access Functions for Elements
    • ElementToSequence(x): GrpGPCElt → [RngIntElt]
    • Eltseq(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
    • IsId(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: Specifying a Polycyclic Presentation
    • Example: Dihedral Group in Different Categories
  • 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: Coercion into a Subgroup
  • 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: Homomorphism of D4 onto Z2 x Z2
    • Example: Direct Product with the Infinite Dihedral Group
• Conversion between Categories
  • AbelianGroup(G): GrpGPC → GrpAb, Map
  • FPGroup(G): GrpGPC → GrpFP, Map
  • PCGroup(G): GrpGPC → GrpPC, Map
  • GPCGroup(G): GrpPerm → GrpGPC, Map
  • GPCGroup(G): GrpMat → GrpGPC, Map
  • GPCGroup(G): GrpAb → GrpGPC, Map
  • GPCGroup(G): GrpPC → GrpGPC, Map
  • Example: Transferring Subgroups and Quotients
• Access Functions for Groups
  • G . i: GrpGPC, RngIntElt → GrpGPCElt
  • Generators(G): GrpGPC → {@ GrpGPCElt @}
  • PCGenerators(G): GrpGPC → {@ GrpGPCElt  @}
  • Generators(H, G): GrpGPC, GrpGPC → {@ GrpGPCElt @}
  • PCGenerators(H, G): GrpGPC, GrpGPC → {@ GrpGPCElt  @}
  • NumberOfGenerators(G): GrpGPC → RngIntElt
  • Ngens(G): GrpGPC → RngIntElt
  • NumberOfPCGenerators(G): GrpGPC → RngIntElt
  • NPCgens(G): GrpGPC → RngIntElt
  • NPCGenerators(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
    • # 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
    • IsSubgroup(H,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
    • Rep(G): GrpGPC → GrpGPCElt
    • RandomProcess(G): GrpGPC → Process
    • Random(P): Process → GrpGPCElt
    • Random(G): GrpGPC → GrpGPCElt
    • Random(G, max): GrpGPC, RngIntElt → GrpGPCElt
• Coset Spaces
  • CosetTable(G, H): GrpGPC, GrpGPC → Map
  • Transversal(G, H): GrpGPC, GrpGPC → {@ GrpGPCElt  @}, Map
  • RightTransversal(G, H): GrpGPC, GrpGPC → {@ GrpGPCElt  @}, Map
  • Example: Right Transversal in the Infinite Dihedral Group
  • CosetAction(G, H): GrpGPC, GrpGPC → Map, GrpPerm, GrpGPC
  • CosetImage(G, H): GrpGPC, GrpGPC → GrpPerm
  • CosetKernel(G, H): GrpGPC, GrpGPC → GrpGPC
  • Example: Coset Action of a Polycyclic Group
• The Subgroup Structure
  • General Subgroup Constructions
    • H ^ g: GrpGPC, GrpGPCElt → GrpGPC
    • Conjugate(H, g): GrpGPC, GrpGPCElt → GrpGPC
    • H ^ G: GrpGPC, GrpGPC → GrpGPC
    • ncl< G | H >: GrpGPC, GrpGPC → GrpGPC
    • NormalClosure(G, H): GrpGPC, GrpGPC → GrpGPC
    • CommutatorSubgroup(G, H, K): GrpGPC, GrpGPC, GrpGPC → GrpGPC
    • CommutatorSubgroup(H, K): GrpGPC, GrpGPC → GrpGPC
  • Subgroup Constructions Requiring a Nilpotent Covering Group
    • H meet K: GrpGPC, GrpGPC → GrpGPC
    • H meet:= K: GrpGPC, GrpGPC → GrpGPC
    • Centraliser(G, g): GrpGPC, GrpGPCElt → GrpGPC
    • Centralizer(G, g): GrpGPC, GrpGPCElt → GrpGPC
    • Centraliser(G, H): GrpGPC, GrpGPC → GrpGPC
    • Centralizer(G, H): GrpGPC, GrpGPC → GrpGPC
    • Core(G, H): GrpGPC, GrpGPC → GrpGPC
    • Normaliser(G, H): GrpGPC, GrpGPC → GrpGPC
    • Normalizer(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
  • IsSolvable(G): GrpGPC → BoolElt
  • General Properties of Subgroups
    • IsCentral(G, H): GrpGPC, GrpGPC → BoolElt
    • IsNormal(G, H): GrpGPC, GrpGPC → BoolElt
  • Properties of Subgroups Requiring a Nilpotent Covering Group
    • IsConjugate(G, H, K): GrpGPC, GrpGPC, GrpGPC → BoolElt, GrpGPCElt
    • IsSelfNormalising(G, H): GrpGPC, GrpGPC → BoolElt
    • IsSelfNormalizing(G, H): GrpGPC, GrpGPC → BoolElt
    • Example: Intersections of Subgroups of a Nilpotent Group
    • Example: Subgroup Structure of a Wreath Product
• Normal Structure and Characteristic Subgroups
  • Characteristic Subgroups and Subgroup Series
    • Centre(G): GrpGPC → GrpGPC
    • Center(G): GrpGPC → GrpGPC
    • DerivedLength(G): GrpGPC → RngIntElt
    • DerivedSeries(G): GrpGPC → [GrpGPC]
    • DerivedSubgroup(G): GrpGPC → GrpGPC
    • DerivedGroup(G): GrpGPC → GrpGPC
    • EFASeries(G): GrpGPC → [GrpGPC]
    • FittingLength(G): GrpGPC → RngIntElt
    • FittingSeries(G): GrpGPC → [GrpGPC]
    • FittingSubgroup(G): GrpGPC → GrpGPC
    • FittingGroup(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: Lower Central Series of a Dihedral Group
  • The Abelian Quotient Structure of a Group
    • AbelianQuotient(G): GrpGPC → GrpAb, Map
    • AbelianQuotientInvariants(G): GrpGPC → [ RngIntElt ]
    • AQInvariants(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: Conjugacy in a Nilpotent Wreath Product
• Representation Theory
  • EFAModuleMaps(G): GrpGPC → [ModGrp]
  • EFAModules(G): GrpGPC → [ModGrp]
  • GModule(G, A, p): GrpGPC, GrpGPC, RngIntElt → ModGrp, Map
  • GModule(G, A): GrpGPC, GrpGPC → ModGrp, Map
  • GModule(G, A, B, p): GrpGPC, GrpGPC, GrpGPC, RngIntElt → ModGrp, Map
  • GModule(G, A, B): GrpGPC, GrpGPC, GrpGPC → 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: Semisimple Modules of a Polycyclic Group
  • Example: Primes of Semisimple G-Modules
  • Example: Fitting Subgroup via Semisimple Modules
  • Example: Refining an EFA-Series with Module Maps
• Power Groups
  • Parent(G): GrpGPC → PowStr
  • PowerGroup(G): GrpPC → PowerGroup
• Bibliography