# ApCoCoA-1:FreeAbelian groups

From ApCoCoAWiki

#### 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 (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press.)

#### 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;