ApCoCoA-1:Higman groups

Higman group

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 LLex has 144 elements</Comment>
 	<Comment>Higman_group</Comment>
 </FREEALGEBRA>