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 (H66E1)
Modifying Limits AutomaticGroup(F: parameters) : GrpFP -> GrpAtc SetVerbose("KBMAG", v) : MonStgElt, RngIntElt -> Example GrpAtc_AutomaticGroup-3 (H66E2) Example GrpAtc_AutomaticGroup-4 (H66E3)
Accessing Group Information G . i : GrpRWS, RngIntElt -> GrpRWSElt Generators(G) : GrpRWS -> [GrpRWSElt] NumberOfGenerators(G) : GrpRWS -> RngIntElt Example GrpAtc_BasicAccess (H66E4)
Properties of an Automatic Group IsFinite(G) : GrpRWS -> BoolElt, RngIntElt Order(G) : GrpRWS -> RngIntElt Example GrpAtc_Order (H66E5) Example GrpAtc_Order-2 (H66E6)
Arithmetic with Words
Construction of a Word G ! [ i1, ..., is ] : GrpAtc, [ RngIntElt ] -> GrpAtcElt Identity(G) : GrpAtc -> GrpAtcElt Parent(w) : GrpAtcElt -> GrpAtc Example GrpAtc_Words (H66E7)
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 (H66E8)
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 (H66E9)
The Growth Function GrowthFunction(G) : GrpAtc -> FldFunRatElt
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