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AUTOMATIC GROUPS

 
Acknowledgements
 
Introduction
      Terminology
      The Category of Automatic Groups
      The Construction of an Automatic Group
 
Creation of Automatic Groups
      Construction of an Automatic Group
      Modifying Limits
      Accessing Group Information
 
Properties of an Automatic Group
 
Arithmetic with Words
      Construction of a Word
      Operations on Elements
 
Homomorphisms
      General Remarks
      Construction of Homomorphisms
 
Set Operations
 
The Growth Function
 
Bibliography







DETAILS

 
Introduction

      Terminology

      The Category of Automatic Groups

      The Construction of an Automatic Group

 
Creation of Automatic Groups

      Construction of an Automatic Group
            AutomaticGroup(F: parameters) : GrpFP -> GrpAtc
            Example GrpAtc_AutomaticGroup (H75E1)

      Modifying Limits
            AutomaticGroup(F: parameters) : GrpFP -> GrpAtc
            SetVerbose("KBMAG", v) : MonStgElt, RngIntElt ->
            Example GrpAtc_AutomaticGroup-3 (H75E2)
            Example GrpAtc_AutomaticGroup-4 (H75E3)

      Accessing Group Information
            G . i : GrpRWS, RngIntElt -> GrpRWSElt
            Generators(G) : GrpRWS -> [GrpRWSElt]
            NumberOfGenerators(G) : GrpRWS -> RngIntElt
            Example GrpAtc_BasicAccess (H75E4)
            FPGroup(G) : GrpAtc -> GrpFP, Map
            WordAcceptor(G) : GrpAtc -> Rec
            WordAcceptorSize(G) : GrpAtc -> RngIntElt, RngIntElt
            WordDifferenceAutomaton(G) : GrpAtc -> Rec
            WordDifferenceSize(G) : GrpAtc -> RngIntElt, RngIntElt
            WordDifferences(G) : GrpAtc -> SeqEnum
            GeneratorOrder(G) : GrpAtc -> SeqEnum

 
Properties of an Automatic Group
      IsFinite(G) : GrpRWS -> BoolElt, RngIntElt
      Order(G) : GrpRWS -> RngIntElt
      Example GrpAtc_Order (H75E5)
      Example GrpAtc_Order-2 (H75E6)

 
Arithmetic with Words

      Construction of a Word
            G ! [ i1, ..., is ] : GrpAtc, [ RngIntElt ] -> GrpAtcElt
            Identity(G) : GrpAtc -> GrpAtcElt
            Parent(w) : GrpAtcElt -> GrpAtc
            Example GrpAtc_Words (H75E7)

      Operations on Elements
            u * v : GrpRWSElt, GrpRWSElt -> GrpRWSElt
            u / v : GrpRWSElt, GrpRWSElt -> GrpRWSElt
            u ^ n : GrpRWSElt, RngIntElt -> GrpRWSElt
            u ^ v : GrpRWSElt, GrpRWSElt -> GrpRWSElt
            Inverse(w) : GrpRWSElt -> GrpRWSElt
            (u, v) : GrpRWSElt, GrpRWSElt -> GrpRWSElt
            (u1, ..., ur) : GrpRWSElt, ..., GrpRWSElt -> GrpRWSElt
            u eq v : GrpRWSElt, GrpRWSElt -> BoolElt
            u ne v : GrpRWSElt, GrpRWSElt -> BoolElt
            IsId(w) : GrpRWSElt -> BoolElt
            # u : GrpRWSElt -> RngIntElt
            ElementToSequence(u) : GrpRWSElt -> [ RngIntElt ]
            Example GrpAtc_Arithmetic (H75E8)

 
Homomorphisms

      General Remarks

      Construction of Homomorphisms
            hom< A -> G | S > : Struct , Struct -> Map

 
Set Operations
      Random(G, n) : GrpAtc, RngIntElt -> GrpAtcElt
      Random(G) : GrpAtc -> GrpAtcElt
      Representative(G) : GrpAtc -> GrpAtcElt
      Set(G, a, b) : GrpAtc, RngIntElt, RngIntElt -> SetEnum
      Set(G) : GrpAtc -> SetEnum
      Seq(G, a, b) : GrpAtc, RngIntElt, RngIntElt -> SeqEnum
      Seq(G) : GrpAtc -> SeqEnum
      Example GrpAtc_Set (H75E9)

 
The Growth Function
      GrowthFunction(G) : GrpAtc -> FldFunRatElt
      Example GrpAtc_GrowthFunction (H75E10)

 
Bibliography

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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012