ApCoCoA-1:Other5 groups

Other groups

Description

The first group, denoted by G, has an order |G| = 4224 and can be represented as:

 G = <a,b | a^{2}b^{-4} = (ababab^{3})^{2} = 1>

The second group, denoted by H, is also solvable and has the following representation:

 H = <a,b | a^{2}b^{4} = (ababab^{3})^{2} = 1>

Reference

No reference available

Computation of G

 /*Use the ApCoCoA package ncpoly.*/
 
 // a is invers to c and b is invers to d
 Use ZZ/(2)[a,b,c,d];
 NC.SetOrdering("LLEX");
 Define CreateRelationsOther5()
   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^{-4} = 1
   Append(Relations,[[a,a,d,d,d,d],[1]]);
   
   // add the relation (ababab^{3})^{2} = 1
   Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b],[1]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther5();
 GB:=NC.GB(Relations,31,1,100,1000);

G in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-24" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>12</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<ncpoly>a*a*d*d*d*d-1</ncpoly>
 	<ncpoly>(a*b*a*b*a*b*b*b)^2-1</ncpoly>
 	<Comment>relation:(ababab^{3})^{2} = 1</Comment>
 	</basis>
 	<Comment>The partial LLex Gb has 96 elements</Comment>
 	<Comment>Other_groups5</Comment>
 </FREEALGEBRA>

Computation of H

 /*Use the ApCoCoA package ncpoly.*/
 
 // a is invers to c and b is invers to d
 Use ZZ/(2)[a,b,c,d];
 NC.SetOrdering("LLEX");
 Define CreateRelationsOther6()
   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^{4} = 1
   Append(Relations,[[a,a,b,b,b,b],[1]]);
   
   // add the relation (ababab^{3})^{2} = 1
   Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b],[1]]);
   
   Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOther6();
 GB:=NC.GB(Relations,31,1,100,1000);

H in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>a,b,c,d</vars>
 	<uptoDeg>13</uptoDeg>
 	<basis>
 	<ncpoly>a*c-1</ncpoly>
 	<ncpoly>c*a-1</ncpoly>
 	<ncpoly>b*d-1</ncpoly>
 	<ncpoly>d*b-1</ncpoly>
 	<Comment>polynomials to define inverse elements</Comment>
 	<ncpoly>a*a*b*b*b*b-1</ncpoly>
 	<ncpoly>(a*b*a*b*a*b*b*b)^2-1</ncpoly>
 	<Comment>relation:(ababab^{3})^{2}= 1</Comment>
 	</basis>
 	<Comment>The LLexGB has 268 elements</Comment>
 	<Comment>Other_groups6</Comment>
 </FREEALGEBRA>