# ApCoCoA-1:Tits group

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### Tits group

#### Description

The Tits group was found by Jacques Tits in 1964. It is a finite simple group and has the following representation:

``` T = <a,b | a^2 = b^3 = (ab)^13 = [a,b]^5 = [a,bab]^4 = ((ab)^4ab^{-1})^6 = 1>
```

#### Reference

Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics. Second Series 80: 313–329

Parrott, David (1972), "A characterization of the Tits' simple group", Canadian Journal of Mathematics 24: 672–685lag

#### Computation

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

Use ZZ/(2)[a,b];
NC.SetOrdering("LLEX");

Define CreateRelationsTits()
Relations:=[];
// add the relation a^2 = 1
Append(Relations,[[a^2],[1]]);

// add the relation b^3 = 1
Append(Relations,[[b^3],[1]]);

// add the relation (ab)^{13} = 1
RelationBuffer1:=[];
For Index1 := 1 To 13 Do
Append(RelationBuffer1, a);
Append(RelationBuffer1, b);
EndFor;
Append(Relations,[RelationBuffer1,[1]]);

// add the relation [a,b]^5 = 1
RelationBuffer2:=[];
// the commutator of [a,b] is a,b,a,b^2
For Index2 := 1 To 5 Do
Append(RelationBuffer2,a);
Append(RelationBuffer2,b);
Append(RelationBuffer2,a);
Append(RelationBuffer2,b^2);
EndFor;
Append(Relations,[RelationBuffer2,[1]]);

// add the relation [a,bab]^4 = 1
RelationBuffer3:=[];
// the commutator of [a,bab] is a,bab,a,b^{2}ab^{2}
For Index2 := 1 To 4 Do
Append(RelationBuffer3,a);
Append(RelationBuffer3,b);
Append(RelationBuffer3,a);
Append(RelationBuffer3,b);
Append(RelationBuffer3,a);
Append(RelationBuffer3,b^2);
Append(RelationBuffer3,a);
Append(RelationBuffer3,b^2);
EndFor;
Append(Relations,[RelationBuffer3,[1]]);

// add the relation ((ab)^4ab^{-1})^6 = 1
RelationBuffer4:=[];
// create (ab)^4
For Index3:= 1 To 4 Do
Append(RelationBuffer4,a);
Append(RelationBuffer4,b);
EndFor;
Append(RelationBuffer4,a);
// b^{-1} = b^2
Append(RelationBuffer4,b^2);

RelationBuffer5:=[];
For Index4:=1 To 6 Do
Foreach Elem In RelationBuffer4 Do
Append(RelationBuffer5,Elem);
EndForeach;
EndFor;
Append(Relations,[RelationBuffer5,[1]]);

Return Relations;
EndDefine;

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