ApCoCoA:Tits group

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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);

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>