Difference between revisions of "ApCoCoA-1:Other2 groups"

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(New page: === <div id="Other2_groups">Other groups</div> === ==== Description ==== The first group is called Rosenberger-Monster and is the largest finite ge...)
 
 
(8 intermediate revisions by 3 users not shown)
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=== <div id="Other2_groups">[[:ApCoCoA:Symbolic data#Other2_groups|Other groups]]</div> ===
+
=== <div id="Other2_groups">[[:ApCoCoA:Symbolic data#Other_groups|Other Groups]]</div> ===
 
==== Description ====
 
==== Description ====
 
The first group is called Rosenberger-Monster and is the largest finite generalized triangle group.
 
The first group is called Rosenberger-Monster and is the largest finite generalized triangle group.
Line 14: Line 14:
 
   /*Use the ApCoCoA package ncpoly.*/
 
   /*Use the ApCoCoA package ncpoly.*/
 
    
 
    
 
 
   Use ZZ/(2)[a,b];
 
   Use ZZ/(2)[a,b];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
 +
 
   Define CreateRelationsOther2()
 
   Define CreateRelationsOther2()
 
     Relations:=[];
 
     Relations:=[];
Line 23: Line 23:
 
     Append(Relations,[[a,a],[1]]);
 
     Append(Relations,[[a,a],[1]]);
 
     Append(Relations,[[b,b,b],[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,b,a,b,b,a,b,a,b,b,a,b,b,a,b,a,b,a,b,a,b,b,a,b,b,a,b,a,b,b,a,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;
 
     Return Relations;
 
   EndDefine;
 
   EndDefine;
 
    
 
    
 
   Relations:=CreateRelationsOther2();
 
   Relations:=CreateRelationsOther2();
   GB:=NC.GB(Relations,31,1,100,1000);
+
   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 ====
 
==== Computation of H ====
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   Use ZZ/(2)[a,b];
 
   Use ZZ/(2)[a,b];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
 +
 
   Define CreateRelationsOther3()
 
   Define CreateRelationsOther3()
 
     Relations:=[];
 
     Relations:=[];
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     Append(Relations,[[a,a],[1]]);
 
     Append(Relations,[[a,a],[1]]);
 
     Append(Relations,[[b,b,b],[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]]);
+
  // 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;
 
     Return Relations;
 
   EndDefine;
 
   EndDefine;
 
    
 
    
 
   Relations:=CreateRelationsOther3();
 
   Relations:=CreateRelationsOther3();
   GB:=NC.GB(Relations,31,1,100,1000);
+
   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>

Latest revision as of 21:08, 22 April 2014

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>