# ApCoCoA-1:SpecialLinear group

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### SpecialLinear group

#### 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:=[];
Append(Relations,[[a,c],]);
Append(Relations,[[c,a],]);
Append(Relations,[[b,d],]);
Append(Relations,[[d,b],]);

// 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],]);

Return Relations;
EndDefine;

Relations:=CreateRelationsSpecialLinear();
GB:=NC.GB(Relations,31,1,100,1000);
```