ApCoCoA-1:Other13 groups

Other Groups

Description

This group has the following finite representation:

G = <x,y | x^2 = xy^{a}xy^{b}xy^{c}xy^{d} = 1>

Reference

No reference available

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 // Note that a,b,c,d >= 1
 MEMORY.A := 3;
 MEMORY.B := 3;
 MEMORY.C := 4;
 MEMORY.D := 5;

 // y is invers to z, the invers element of x follows directly from the relation x^2 = 1
 Use ZZ/(2)[x,y,z];
 NC.SetOrdering("LLEX");
 Define CreateRelationsOther13()
   Relations:=[];
   
   // add the relation of the invers elements yz = zy = 1
   Append(Relations,[[y,z],[1]]);
   Append(Relations,[[z,y],[1]]);
   
   // add the relation x^2 = 1
   Append(Relations,[[x,x],[1]]);
   
   // add the relation xy^{a}xy^{b}xy^{c}xy^{d}
   Append(Relations,[[x,y^(MEMORY.A),x,y^(MEMORY.B),x,y^(MEMORY.C),x,y^(MEMORY.D)],[1]]);

   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther13();
 Gb:=NC.GB(Relations,31,1,100,1000);

Examples in Symbolic Data Format

Other group 13 a=2 b=3 c=3 d=4

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<uptoDeg>19</uptoDeg>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>x*(y^2)*x*(y^3)*x*(y^3)*x*(y^4)-1</ncpoly>
 	<ncpoly></ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 198 elements</Comment>
 	<Comment>Other_groups_13a2b3c3d4</Comment>
 </FREEALGEBRA>

Other group 13 a=3 b=3 c=4 d=5

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<uptoDeg>18</uptoDeg>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>x*(y^3)*x*(y^3)*x*(y^4)*x*(y^5)-1</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 4 elements</Comment>
 	<Comment>Other_groups_13a3b3c4d5</Comment>
 </FREEALGEBRA>

Other group 13 a=5 b=2 c=4 d=3

 <FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier">
 	<vars>x,y,z</vars>
 	<uptoDeg>17</uptoDeg>
 	<basis>
 	<ncpoly>y*z-1</ncpoly>
 	<ncpoly>z*y-1</ncpoly>
 	<ncpoly>x*x-1</ncpoly>
 	<ncpoly>x*(y^5)*x*(y^2)*x*(y^4)*x*(y^3)-1</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 4 elements</Comment>
 	<Comment>Other_groups_13a5b2c4d3</Comment>
 </FREEALGEBRA>