Difference between revisions of "ApCoCoA-1:Free groups"
From ApCoCoAWiki
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free group has a unique representation. | free group has a unique representation. | ||
F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1> | 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 ==== | ==== Computation ==== | ||
Revision as of 09:17, 23 August 2013
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;
