Free Groups

• Introduction
• Free Groups and Words
  • Construction of a Free Group
    • FreeGroup(n): RngIntElt → GrpFP
    • Example: Free
  • Construction of Words
    • G ! [ i₁, ..., iₛ ]: GrpFP, [ RngIntElt ] → GrpFPElt
    • Identity(G): GrpFP → GrpFPElt
    • Id(G): GrpFP → GrpFPElt
    • G ! 1: GrpFP, RngIntElt → GrpFPElt
    • Random(G, m, n): GrpFP, RngIntElt, RngIntElt → GrpFPElt
  • Access Functions for Words
    • # w: GrpFPElt → RngIntElt
    • ElementToSequence(w): GrpFPElt → [ RngIntElt ]
    • Eltseq(w): GrpFPElt → [ RngIntElt ]
    • ExponentSum(w, x): GrpFPElt, GrpFPElt → RngIntElt
    • Weight(w, x): GrpFPElt, GrpFPElt → RngIntElt
    • GeneratorNumber(w): GrpFPElt → RngIntElt
    • LeadingGenerator(w): GrpFPElt → GrpFPElt
    • Parent(w): GrpFPElt → GrpFP
    • Example: Word Access
  • Arithmetic Operators for Words
    • u * v: GrpFPElt, GrpFPElt → GrpFPElt
    • u ^ n: GrpFPElt, RngIntElt → GrpFPElt
    • u ^ v: GrpFPElt, GrpFPElt → GrpFPElt
    • (u, v): GrpFPElt, GrpFPElt → GrpFPElt
    • (u₁, ..., uₙ): List(GrpFPElt) → GrpFPElt
  • Comparison of Words
    • u eq v: GrpFPElt, GrpFPElt → BoolElt
    • u ne v: GrpFPElt, GrpFPElt → BoolElt
    • u lt v: GrpFPElt, GrpFPElt → BoolElt
    • u le v: GrpFPElt, GrpFPElt → BoolElt
    • u ge v: GrpFPElt, GrpFPElt → BoolElt
    • u gt v: GrpFPElt, GrpFPElt → BoolElt
    • Example: Words
  • String Operations on Words
    • Eliminate(u, x, v): GrpFPElt, GrpFPElt, GrpFPElt → GrpFPElt
    • Eliminate(U, x, v): { GrpFPElt }, GrpFPElt, GrpFPElt → { GrpFPElt }
    • Match(u, v, f): GrpFPElt, GrpFPElt, RngIntElt → BoolElt, RngIntElt
    • RotateWord(u, n): GrpFPElt, RngIntElt → GrpFPElt
    • Substitute(u, f, n, v): GrpFPElt, RngIntElt, RngIntElt, GrpFPElt → GrpFPElt
    • Subword(u, f, n): GrpFPElt, RngIntElt, RngIntElt → GrpFPElt
    • Example: Word Ops
• Finitely Generated Subgroups of Free Groups
  • Supergroup(F): GrpFP → GrpFP
  • x in H: GrpFP, GrpFPElt → BoolElt
  • IsSubgroup(H, K): GrpFP, GrpFP → BoolElt
  • H eq K: GrpFP, GrpFP → BoolElt
  • Index(F, H): GrpFP, GrpFP → RngIntElt
  • HasFiniteIndex(F, H): GrpFP, GrpFP → BoolElt
  • FreeGenerators(H): GrpFP → SeqEnum, GrpFP
  • H meet K: GrpFP, GrpFP → GrpFP
  • Centraliser(F,x): GrpFP, GrpFPElt → GrpFP
  • Centralizer(F,x): GrpFP, GrpFPElt → GrpFP
  • IsConjugate(F, x, y): GrpFP, GrpFPElt, GrpFPElt → BoolElt, GrpFPElt
  • Centraliser(F, H): GrpFP, GrpFP → GrpFP
  • Centralizer(F, H): GrpFP, GrpFP → GrpFP
  • Normaliser(F, H): GrpFP, GrpFP → GrpFP
  • Normalizer(F, H): GrpFP, GrpFP → GrpFP
  • IsConjugate(F, H, K): GrpFP, GrpFP, GrpFP → BoolElt, GrpFPElt
  • Example: Free Subgroups
• The Automorphism Group of a Free Group
  • AutomorphismGroup(F): GrpFP → GrpAuto
  • InverseAutomorphismFreeGroup(F, Q): GrpFP, SeqEnum → GrpAutoElt
  • WhiteheadReduction(F, Q): GrpFP, SeqEnum → BoolElt, SeqEnum, GrpAutoElt
  • Example: Free Aut
  • Example: Free Aut2