# Difference between revisions of "ApCoCoA-1:Tits group"

### 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],]);

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

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

// 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,]);

// 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,]);

// 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,]);

Return Relations;
EndDefine;

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

#### Example in Symbolic Data Format

``` <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
<vars>a,b</vars>
<basis>
<ncpoly>a*a-1</ncpoly>
<ncpoly>b*b*b-1</ncpoly>
<ncpoly>((a*b)^13)-1</ncpoly>
<ncpoly>((a*b*a*b*b)^5)-1</ncpoly>
<ncpoly>(a*b*a*b*a*b^2*a*b^2)^4-1</ncpoly>
<ncpoly>(a*b*a*b*a*b*a*b*a*b^2)^6-1</ncpoly>
</basis>
<Comment>Tits_group</Comment>
</FREEALGEBRA>
```