ApCoCoA-1:Other1 groups
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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 partial LLex Gb has 137 elements</Comment> <Comment>Other_groups1k63</Comment> </FREEALGEBRA>