ApCoCoA-1:Free groups

Free groups

Description

The relations of a free group with n generators only consists of the existence of the invers elements. Any element of a free group has a unique representation.

 F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1>

Reference

Kharlampovich, Olga; Myasnikov, Alexei, "Elementary theory of free non-abelian groups". J. Algebra 302 (2): 451–552, 2006.

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Number of free group
 MEMORY.N:=4;
 
 Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsFree()
   Relations:=[];
   For Index1 := 1 To MEMORY.N Do
     Append(Relations,[[x[Index1],y[Index1]],[1]]);
     Append(Relations,[[y[Index1],x[Index1]],[1]]);
   EndFor;
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsFree();
 Relations;
 
 Gb:=NC.GB(Relations);
 Gb;