ApCoCoA-1:Free groups: Difference between revisions
From ApCoCoAWiki
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/*Use the ApCoCoA package ncpoly.*/ | /*Use the ApCoCoA package ncpoly.*/ | ||
// Number of free | // Number of free group | ||
MEMORY.N:=4; | MEMORY.N:=4; | ||
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Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
Define | Define CreateRelationsFree() | ||
Relations:=[]; | Relations:=[]; | ||
For Index1 := 1 To MEMORY.N Do | For Index1 := 1 To MEMORY.N Do | ||
| Line 24: | Line 24: | ||
EndDefine; | EndDefine; | ||
Relations:= | Relations:=CreateRelationsFree(); | ||
Relations; | Relations; | ||
GB:=NC.GB(Relations); | GB:=NC.GB(Relations); | ||
GB; | GB; | ||
Revision as of 08:26, 16 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 (2006). "Elementary theory of free non-abelian groups". J. Algebra 302 (2): 451–552)
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;
