# 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;