ApCoCoA-1:SpecialLinearPrime group

Special Linear Group with Prime p

Description

For p is prime has the Special Linear Group with prime p the following presentation:

```  SL_2(p) = <x,y | x^{2} =(xy)^{3},(xy^{4}xy^{t})^{2}y^{p}x^{2k}=1>
```

Computation

``` /*Use the ApCoCoA package ncpoly.*/

// set the variables k,p,t
// Note that p have to be prime
MEMORY.K:=3;
MEMORY.P:=2;
MEMORY.T:=5;
Use ZZ/(2)[x,y,a,b];
NC.SetOrdering("LLEX");

Define CreateRelationsSpeciallineargroupprime()
Relations:=[];

// add the inverse relations
Append(Relations,[[x,a],[1]]);
Append(Relations,[[a,x],[1]]);
Append(Relations,[[y,b],[1]]);
Append(Relations,[[b,y],[1]]);

// add the relation x^2 = (xy)^3
Append(Relations,[[x,x],[x,y,x,y,x,y]]);

// add the relation ((((x*y)^4)*x*y^t)^2)*(y^p)*(x^(2k))-1
Append(Relations,[[x,y,x,y,x,y,x,y,x,y^(MEMORY.T),x,y,x,y,x,y,x,y,x,y^(MEMORY.T),y^(MEMORY.P),x^(2*MEMORY.K)],[1]]);
Return Relations;
EndDefine;

Relations:=CreateRelationsSpeciallineargroupprime();
Relations;

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

Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
<vars>x,y,a,b</vars>
<uptoDeg>12</uptoDeg>
<basis>
<ncpoly>x*x-(x*y)^3</ncpoly>
<ncpoly>((((x*y)^4)*x*y^5)^2)*(y^2)*(x^(2*3))-1</ncpoly>
</basis>
<Comment>The partial LLEX Gb has 285 elements</Comment>
<Comment>Special_Linear_group with prime p_k3p2t5</Comment>
</FREEALGEBRA>
```