ApCoCoA-1:Other1 groups

Other groups

Description

This group has the following representation:

 G = <a,b | a^{2}b^{-6} = (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1>

where k is congruent to 3 mod 6.

Reference

No reference available

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 //K is congruent to 3 mod 6
 MEMORY.K:=3;
 // a is invers to c and b is invers to d
 Use ZZ/(2)[a,b,c,d];
 NC.SetOrdering("LLEX");
 Define CreateRelationsOther1()
   Relations:=[];
   
   // add the invers relations ac = ca = bd = db = 1
   Append(Relations,[[a,c],[1]]);
   Append(Relations,[[c,a],[1]]);
   Append(Relations,[[b,d],[1]]);
   Append(Relations,[[d,b],[1]]);
   
   // add the relation a^{2}b^{-6} = aadddddd = 1
   Append(Relations,[[a,a,d,d,d,d,d,d],[1]]);
   
   // add the relation (ab^{-1})^{3}ab^{-2}ab^{k}a^{-1}b = 1
   Append(Relations,[[a,d,a,d,a,d,a,d,d,a,b^MEMORY.K,c,b],[1]]);
 
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther1();
 Gb:=NC.GB(Relations,31,1,100,1000);

Examples in Symbolic Data Format

Other group 1 k=3

 <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>10</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>(a^2)*(d^6)-1</ncpoly>
 	<ncpoly>((a*d)^3)*a*d*d*a*(b^3)*c*b-1</ncpoly>
 	</basis>
 	<Comment>Other_groups1k3</Comment>
 </FREEALGEBRA>

Other group 1 k=39

 <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>10</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>(a^2)*(d^6)-1</ncpoly>
 	<ncpoly>((a*d)^3)*a*d*d*a*(b^39)*c*b-1</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 234 elements</Comment>
 	<Comment>Other_groups1k39</Comment>
 </FREEALGEBRA>

Other group 1 k=63

 <FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>9</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>(a^2)*(d^6)-1</ncpoly>
 	<ncpoly>((a*d)^3)*a*d*d*a*(b^63)*c*b-1</ncpoly>
 	</basis>
 	<Comment>The LLexGB has 137 elements</Comment>
 	<Comment>Other_groups1k63</Comment>
 </FREEALGEBRA>