# ApCoCoA-1:SpecialLinear group

From ApCoCoAWiki

#### Description

The special linear group over Z is the group of all n x n matrices with coefficients in Z which has a determinant equals 0. A simple representation is given by:

SL_2(Z) = <a,b | aba = bab, (aba)^4 = 1 >

#### Reference

Andrew Baker: An introduction to matrix groups and their applications

#### Computation

/*Use the ApCoCoA package ncpoly.*/ // a is invers to c, b is invers to d Use ZZ/(2)[a,b,c,d]; NC.SetOrdering("LLEX"); Define CreateRelationsSpecialLinear() Relations:=[]; // add the invers relations Append(Relations,[[a,c],[1]]); Append(Relations,[[c,a],[1]]); Append(Relations,[[b,d],[1]]); Append(Relations,[[d,b],[1]]); // add the relation aba = bab Append(Relations,[[a,b,a],[b,a,b]]); // add the relation (aba)^4 = 1 Append(Relations,[[a,b,a,a,b,a,a,b,a,a,b,a],[1]]); Return Relations; EndDefine; Relations:=CreateRelationsSpecialLinear(); GB:=NC.GB(Relations,31,1,100,1000);