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);

G in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-02-27" createdBy="strohmeier">
 	<vars>a,b</vars>
 	<uptoDeg>39</uptoDeg>
 	<basis>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>b*b*b-1</ncpoly>
 	<Comment>polynomials to define inverse elements</Comment>
 	<ncpoly>(a*b*a*b*a*b*a*b*b*a*b*b*a*b*a*b*b*a*b*b)^2-1</ncpoly>
 	<Comment>(abababab^{2}ab^{2}abab^{2}ab^{2})^2 = 1 </Comment>
 	</basis>
 	<Comment> The partial LLex Gb has 3 elements</Comment>
 	<Comment>Other_groups2</Comment>
 </FREEALGEBRA>

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^2,a,b,a,b^2,a,b^2,a,b,a,b,a,b,a,b^2,a,b,a,b^2,a,b^2],[1]]);

   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther3();
 Gb:=NC.GB(Relations,31,1,100,1000);

H in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-02-27" createdBy="strohmeier">
 	<vars>a,b</vars>
 	<uptoDeg>34</uptoDeg>
 	<basis>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>b*b*b-1</ncpoly>
 	<ncpoly>(a*b*a*b*a*b*a*b*b*a*b*a*b*b*a*b*b)^2-1</ncpoly>
 	<Comment>relation: (abababab^{2}abab^{2}ab^{2})^2 = 1</Comment>
 	</basis>
 	<Comment>The partial LLex Gb has 249 elements</Comment>
 	<Comment>Other_groups3</Comment>
 </FREEALGEBRA>