ApCoCoA-1:Other2 groups

Other groups

Description

The first group is called Rosenberger-Monster and is the largest finite generalized triangle group. A finite representation of G is given below:

 G = <a,b | a^2 = b^3 = (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1>

The second group is already infinite and denoted by H:

 H = <a,b | a^2 = b^3 = (abababab^{2}abab^{2}ab^{2})^2 = 1>

Reference

On the Rosenberger Monster Robert Fitzgerald Morse, Department of Electrical Engineering and Computer Science, University of Evansville IN 47722 USA

Computation of G

 /*Use the ApCoCoA package ncpoly.*/
 
 Use ZZ/(2)[a,b];
 NC.SetOrdering("LLEX");

 Define CreateRelationsOther2()
   Relations:=[];
   
   // add the relations a^2 = b^3 = 1
   Append(Relations,[[a,a],[1]]);
   Append(Relations,[[b,b,b],[1]]);
 
  // add the relation (abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1
   Append(Relations,[[a,b,a,b,a,b,a,b^2,a,b^2,a,b,a,b^2,a,b^2,a,b,a,b,a,b,a,b^2,a,b^2,a,b,a,b^2,a,b^2],[1]]);

   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther2();
 GB:=NC.GB(Relations,31,1,100,1000);

Computation of H

 /*Use the ApCoCoA package ncpoly.*/
 
 Use ZZ/(2)[a,b];
 NC.SetOrdering("LLEX");
 Define CreateRelationsOther3()
   Relations:=[];
   
   // add the relations a^2 = b^3 = 1
   Append(Relations,[[a,a],[1]]);
   Append(Relations,[[b,b,b],[1]]);
   // add the relation (abababab^{2}abab^{2}ab^{2})^2 = 1
   Append(Relations,[[a,b,a,b,a,b,a,b,b,a,b,a,b,b,a,b,b,a,b,a,b,a,b,a,b,b,a,b,a,b,b,a,b,b],[1]]);
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther3();
 GB:=NC.GB(Relations,31,1,100,1000);