ApCoCoA-1: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>