ApCoCoA-1:Other12 groups
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Description
This group has the following finite representation:
G = <x,t | tx^{a}t^{-1} = x^{b},t^{n} = 1>
for a,b >= 1 and n >= 2.
Reference
No reference available
Computation
/*Use the ApCoCoA package ncpoly.*/ // Note that a,b >= 1 and n >= 2 MEMORY.A := 3; MEMORY.B := 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 CreateRelationsOther12() 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 tx^{a}t^{-1} = x^{b} RelationBuffer1:=[]; Append(RelationBuffer1,t); Append(RelationBuffer1,x^(MEMORY.A)); Append(RelationBuffer1,t^(MEMORY.N-1)); Append(Relations,[RelationBuffer1,[x^MEMORY.B]]); Return Relations; EndDefine; Relations:=CreateRelationsOther12(); Gb:=NC.GB(Relations,31,1,100,1000);
Examples in Symbolic Data Format
Other group 12 a=1 b=1 n=2
<FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> <vars>t,x,z</vars> <basis> <ncpoly>x*z-1</ncpoly> <ncpoly>z*x-1</ncpoly> <ncpoly>(t^2)-1</ncpoly> <ncpoly>t*(x^1)*(t^(2-1))-(x^1)</ncpoly> </basis> <Comment>Other_groups_12a1b1n2</Comment> </FREEALGEBRA>
Other group 12 a=3 b=3 n=4
<FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> <vars>t,x,z</vars> <uptoDeg>10</uptoDeg> <basis> <ncpoly>x*z-1</ncpoly> <ncpoly>z*x-1</ncpoly> <ncpoly>(t^4)-1</ncpoly> <ncpoly>t*(x^3)*(t^(4-1))-(x^3)</ncpoly> </basis> <Comment>The partial LLex Gb has 106 elements</Comment> <Comment>Other_groups_12a3b3n4</Comment> </FREEALGEBRA>
Other group 12 a=4 b=4 n=4
<FREEALGEBRA createdAt="2014-01-27" createdBy="strohmeier"> <vars>t,x,z</vars> <uptoDeg>11</uptoDeg> <basis> <ncpoly>x*z-1</ncpoly> <ncpoly>z*x-1</ncpoly> <ncpoly>(t^4)-1</ncpoly> <ncpoly>t*(x^4)*(t^(4-1))-(x^4)</ncpoly> </basis> <Comment>The partial LLex Gb has 182 elements</Comment> <Comment>Other_groups_12a4b4n4</Comment> </FREEALGEBRA>