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

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=== <div id="Higman_groups">[[:ApCoCoA:Symbolic data#Higman_group|Higman group]]</div> ===
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=== <div id="Higman_groups">[[:ApCoCoA:Symbolic data#Higman_group|Higman Group]]</div> ===
 
==== Description ====
 
==== Description ====
 
The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the
 
The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the
 
elements a,b,c and d with the following relations:
 
elements a,b,c and d with the following relations:
 
   H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>
 
   H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>
(Reference: Higman, Graham (1951), "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64)
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 +
==== Reference ====
 +
Graham Higman, "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64, 1951.
  
 
==== Computation ====
 
==== Computation ====
 
   /*Use the ApCoCoA package ncpoly.*/
 
   /*Use the ApCoCoA package ncpoly.*/
 
    
 
    
   // a is invers to e, b is invers to f, c is invers to g and d is invers to h
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   // a is inverse to e, b is inverse to f, c is inverse to g and d is inverse to h
 
   Use ZZ/(2)[a,b,c,d,e,f,g,h];
 
   Use ZZ/(2)[a,b,c,d,e,f,g,h];
 
   NC.SetOrdering("LLEX");
 
   NC.SetOrdering("LLEX");
 +
 
 
   Define CreateRelationsHigman()
 
   Define CreateRelationsHigman()
 
     Relations:=[];
 
     Relations:=[];
 
      
 
      
     // add the relation of the invers element
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     // add the relations of the inverse elements
 
     Append(Relations,[[a,e],[1]]);
 
     Append(Relations,[[a,e],[1]]);
 
     Append(Relations,[[e,a],[1]]);
 
     Append(Relations,[[e,a],[1]]);
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   Relations:=CreateRelationsHigman();
 
   Relations:=CreateRelationsHigman();
 
   Relations;
 
   Relations;
   GB:=NC.GB(Relations,31,1,100,1000);
+
    
   GB;
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  Gb:=NC.GB(Relations,31,1,100,1000);
   Size(GB);
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   Gb;
 +
   Size(Gb);
 +
 
 +
====Example in Symbolic Data Format====
 +
  <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
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  <vars>a,b,c,d,e,f,g,h</vars>
 +
  <uptoDeg>5</uptoDeg>
 +
  <basis>
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  <ncpoly>a*e-1</ncpoly>
 +
  <ncpoly>e*a-1</ncpoly>
 +
  <ncpoly>b*f-1</ncpoly>
 +
  <ncpoly>f*b-1</ncpoly>
 +
  <ncpoly>c*g-1</ncpoly>
 +
  <ncpoly>g*c-1</ncpoly>
 +
  <ncpoly>d*h-1</ncpoly>
 +
  <ncpoly>h*d-1</ncpoly>
 +
  <ncpoly>e*b*a-b*b</ncpoly>
 +
  <ncpoly>f*c*b-c*c</ncpoly>
 +
  <ncpoly>g*d*c-d*d</ncpoly>
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  <ncpoly>h*a*d-a*a</ncpoly>
 +
  </basis>
 +
  <Comment>The partial LLex Gb has 144 elements</Comment>
 +
  <Comment>Higman_group</Comment>
 +
  </FREEALGEBRA>

Latest revision as of 20:50, 22 April 2014

Description

The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the elements a,b,c and d with the following relations:

 H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>

Reference

Graham Higman, "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64, 1951.

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // a is inverse to e, b is inverse to f, c is inverse to g and d is inverse to h
 Use ZZ/(2)[a,b,c,d,e,f,g,h];
 NC.SetOrdering("LLEX");
 
 Define CreateRelationsHigman()
   Relations:=[];
   
   // add the relations of the inverse elements
   Append(Relations,[[a,e],[1]]);
   Append(Relations,[[e,a],[1]]);
   Append(Relations,[[b,f],[1]]);
   Append(Relations,[[f,b],[1]]);
   Append(Relations,[[c,g],[1]]);
   Append(Relations,[[g,c],[1]]);
   Append(Relations,[[d,h],[1]]);
   Append(Relations,[[h,d],[1]]);
   
   // add the relation a^{-1}ba = b^2
   Append(Relations,[[e,b,a],[b^2]]);
   
   // add the relation b^{-1}cb = c^2
   Append(Relations,[[f,c,b],[c^2]]);
   
   // add the relation c^{-1}dc = d^2
   Append(Relations, [[g,d,c],[d^2]]);
   
   // add the relation d^{-1}ad = a^2
   Append(Relations, [[h,a,d],[a^2]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsHigman();
 Relations;
 
 Gb:=NC.GB(Relations,31,1,100,1000);
 Gb;
 Size(Gb);

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>a,b,c,d,e,f,g,h</vars>
 	<uptoDeg>5</uptoDeg>
 	<basis>
 	<ncpoly>a*e-1</ncpoly>
 	<ncpoly>e*a-1</ncpoly>
 	<ncpoly>b*f-1</ncpoly>
 	<ncpoly>f*b-1</ncpoly>
 	<ncpoly>c*g-1</ncpoly>
 	<ncpoly>g*c-1</ncpoly>
 	<ncpoly>d*h-1</ncpoly>
 	<ncpoly>h*d-1</ncpoly>
 	<ncpoly>e*b*a-b*b</ncpoly>
 	<ncpoly>f*c*b-c*c</ncpoly>
 	<ncpoly>g*d*c-d*d</ncpoly>
 	<ncpoly>h*a*d-a*a</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 144 elements</Comment>
 	<Comment>Higman_group</Comment>
 </FREEALGEBRA>