# ApCoCoA-1:SpecialLinear group

### Special Linear 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);
```

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
<vars>a,b,c,d</vars>
<uptoDeg>9</uptoDeg>
<basis>
<ncpoly>a*c-1</ncpoly>
<ncpoly>c*a-1</ncpoly>
<ncpoly>b*d-1</ncpoly>
<ncpoly>d*b-1</ncpoly>
<ncpoly>a*b*a-b*a*b</ncpoly>
<ncpoly>(a*b*a)^4-1</ncpoly>
</basis>
<Comment>Special_Linear_group</Comment>
</FREEALGEBRA>
```