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

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− | === <div id="FreeAbelian_groups">[[:ApCoCoA:Symbolic data#Free_abelian_group|Free | + | === <div id="FreeAbelian_groups">[[:ApCoCoA:Symbolic data#Free_abelian_group|Free Abelian Groups]]</div> === |

==== Description ==== | ==== Description ==== | ||

Every element in a free abelian group can be written in only way as a finite linear combination. A representation is given by the | Every element in a free abelian group can be written in only way as a finite linear combination. A representation is given by the |

## Revision as of 20:47, 22 April 2014

#### Description

Every element in a free abelian group can be written in only way as a finite linear combination. A representation is given by the following:

Z(n) = <a_{1},...,a_{n} | [a_{i},a_{j}] = 1 for all i,j>

#### Reference

Phillip A. Griffith, Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press, 1970.

#### Computation

/*Use the ApCoCoA package ncpoly.*/ // Number of free abelian group MEMORY.N:=3; Use ZZ/(2)[x[1..MEMORY.N],y[1..MEMORY.N]]; NC.SetOrdering("LLEX"); Define CreateRelationsFreeAbelian() Relations:=[]; For Index1 := 1 To MEMORY.N Do For Index2 := 1 To MEMORY.N Do Append(Relations,[[x[Index1],x[Index2],y[Index1],y[Index2]],[1]]); EndFor; EndFor; Return Relations; EndDefine; Relations:=CreateRelationsFreeAbelian(); Relations; Gb:=NC.GB(Relations); Gb;