ApCoCoA-1:Other11 groups
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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>