ApCoCoA-1:Other12 groups

Other Groups

Description

This group has the following finite representation:

G = <x,t | tx^{a}t^{-1} = x^{b},t^{n} = 1>

for a,b >= 1 and n >= 2.

Reference

No reference available

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Note that a,b >= 1 and n >= 2
 MEMORY.A := 3;
 MEMORY.B := 3;
 MEMORY.N := 4;

 // x is invers to z, t has an implicit invers (Relation: t^{n} = 1)
 Use ZZ/(2)[x,t,z];
 NC.SetOrdering("LLEX");

 Define CreateRelationsOther12()
   Relations:=[];
   
   // add the invers relations xz = zx = 1
   Append(Relations,[[x,z],[1]]);
   Append(Relations,[[z,x],[1]]);
   
   // add the relation t^{n} = 1
   RelationBuffer0:=[];
   For Index0:=1 To MEMORY.N Do
     Append(RelationBuffer0,t);
   EndFor;
   Append(Relations,[RelationBuffer0,[1]]);
   
   // add the relation tx^{a}t^{-1} = x^{b}
   RelationBuffer1:=[];
   Append(RelationBuffer1,t);
   Append(RelationBuffer1,x^(MEMORY.A));
   Append(RelationBuffer1,t^(MEMORY.N-1));
   Append(Relations,[RelationBuffer1,[x^MEMORY.B]]);

   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther12();
 Gb:=NC.GB(Relations,31,1,100,1000);

Examples in Symbolic Data Format

Other group 12 a=1 b=1 n=2

 <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^2)-1</ncpoly>
 	<ncpoly>t*(x^1)*(t^(2-1))-(x^1)</ncpoly>
 	</basis>
 	<Comment>Other_groups_12a1b1n2</Comment>
 </FREEALGEBRA>

Other group 12 a=3 b=3 n=4

 <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<uptoDeg>10</uptoDeg>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^4)-1</ncpoly>
 	<ncpoly>t*(x^3)*(t^(4-1))-(x^3)</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 106 elements</Comment>
 	<Comment>Other_groups_12a3b3n4</Comment>
 </FREEALGEBRA>

Other group 12 a=4 b=4 n=4

 <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<uptoDeg>11</uptoDeg>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^4)-1</ncpoly>
 	<ncpoly>t*(x^4)*(t^(4-1))-(x^4)</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 182 elements</Comment>
 	<Comment>Other_groups_12a4b4n4</Comment>
 </FREEALGEBRA>