# Difference between revisions of "ApCoCoA-1:Free groups"

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(New page: === <div id="Free_groups">Free groups</div> === ==== Description ==== F(n) = <a_{1},...,a_{n} | a_{i}a_{i}^{-1} = a_{i}^{-1}a_{i} = 1> (Reference:...) |
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=== <div id="Free_groups">[[:ApCoCoA:Symbolic data#Free_groups|Free groups]]</div> === | === <div id="Free_groups">[[:ApCoCoA:Symbolic data#Free_groups|Free groups]]</div> === | ||

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

## Revision as of 08:15, 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 CreateRelationsFreeAbelian() 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:=CreateRelationsFreeAbelian(); Relations; GB:=NC.GB(Relations); GB;