ApCoCoA-1:Other11 groups

Other Groups

Description

This group has the following finite representation:

G = <x,t | xt^{r} = tx^{r},t^{n} = 1>

for r >= 1 and n >= 2

Reference

No reference available

Computation of G

 /*Use the ApCoCoA package ncpoly.*/
 
 // Note that r >= 1 and n >= 2
 MEMORY.R := 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 CreateRelationsOther11()
   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 xt^{r} = tx^{r}
   RelationBuffer1:=[];
   RelationBuffer2:=[];
   Append(RelationBuffer1,x);
   Append(RelationBuffer2,t);
   For Index1:= 1 To MEMORY.R Do
     Append(RelationBuffer1,t);
     Append(RelationBuffer2,x);
   EndFor;
   Append(Relations,[RelationBuffer1,RelationBuffer2]);
 
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther11();
 Gb:=NC.GB(Relations,31,1,100,1000);

Examples in Symbolic Data Format

Other group 11 r=3 n=4

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<uptoDeg>13</uptoDeg>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^4)-1</ncpoly>
 	<ncpoly>(x*(t^3))-t*(x^3)</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 248 elements</Comment>
 	<Comment>Other_groups_11_r3n4</Comment>
 </FREEALGEBRA>

Other group 11 r=5 n=5

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<uptoDeg>100</uptoDeg>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^5)-1</ncpoly>
 	<ncpoly>(x*(t^5))-t*(x^5)</ncpoly>
 	</basis>
 	<Comment>Other_groups_11_r5n5</Comment>
 </FREEALGEBRA>

Other group 11 r=6 n=7

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<uptoDeg>12</uptoDeg>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^7)-1</ncpoly>
 	<ncpoly>(x*(t^6))-t*(x^6)</ncpoly>
 	</basis>
 	<Comment> The partial LLex Gb has 217 elements </Comment>
 	<Comment>Other_groups_11_r6n7</Comment>
 </FREEALGEBRA>

Other group 11 r=7 n=11

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>t,x,z</vars>
 	<uptoDeg>19</uptoDeg>
 	<basis>
 	<ncpoly>x*z-1</ncpoly>
 	<ncpoly>z*x-1</ncpoly>
 	<ncpoly>(t^11)-1</ncpoly>
 	<ncpoly>(x*(t^7))-t*(x^7)</ncpoly>
 	</basis>
 	<Comment> The partial LLex Gb has 228 elements </Comment>
 	<Comment>Other_groups_11_r7n11</Comment>	
 </FREEALGEBRA>