ApCoCoA-1:Other13 groups
From ApCoCoAWiki
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
Description
This group has the following finite representation:
G = <x,y | x^2 = xy^{a}xy^{b}xy^{c}xy^{d} = 1>
Reference
No reference available
Computation
/*Use the ApCoCoA package ncpoly.*/ // Note that a,b,c,d >= 1 MEMORY.A := 3; MEMORY.B := 3; MEMORY.C := 4; MEMORY.D := 5; // y is invers to z, the invers element of x follows directly from the relation x^2 = 1 Use ZZ/(2)[x,y,z]; NC.SetOrdering("LLEX"); Define CreateRelationsOther13() Relations:=[]; // add the relation of the invers elements yz = zy = 1 Append(Relations,[[y,z],[1]]); Append(Relations,[[z,y],[1]]); // add the relation x^2 = 1 Append(Relations,[[x,x],[1]]); // add the relation xy^{a}xy^{b}xy^{c}xy^{d} Append(Relations,[[x,y^(MEMORY.A),x,y^(MEMORY.B),x,y^(MEMORY.C),x,y^(MEMORY.D)],[1]]); Return Relations; EndDefine; Relations:=CreateRelationsOther13(); Gb:=NC.GB(Relations,31,1,100,1000);
Examples in Symbolic Data Format
Other group 13 a=2 b=3 c=3 d=4
<FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier"> <vars>x,y,z</vars> <uptoDeg>19</uptoDeg> <basis> <ncpoly>y*z-1</ncpoly> <ncpoly>z*y-1</ncpoly> <ncpoly>x*x-1</ncpoly> <ncpoly>x*(y^2)*x*(y^3)*x*(y^3)*x*(y^4)-1</ncpoly> <ncpoly></ncpoly> </basis> <Comment>The partial LLex Gb has 198 elements</Comment> <Comment>Other_groups_13a2b3c3d4</Comment> </FREEALGEBRA>
Other group 13 a=3 b=3 c=4 d=5
<FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier"> <vars>x,y,z</vars> <uptoDeg>18</uptoDeg> <basis> <ncpoly>y*z-1</ncpoly> <ncpoly>z*y-1</ncpoly> <ncpoly>x*x-1</ncpoly> <ncpoly>x*(y^3)*x*(y^3)*x*(y^4)*x*(y^5)-1</ncpoly> </basis> <Comment>The partial LLex Gb has 4 elements</Comment> <Comment>Other_groups_13a3b3c4d5</Comment> </FREEALGEBRA>
Other group 13 a=5 b=2 c=4 d=3
<FREEALGEBRA createdAt="2014-03-04" createdBy="strohmeier"> <vars>x,y,z</vars> <uptoDeg>17</uptoDeg> <basis> <ncpoly>y*z-1</ncpoly> <ncpoly>z*y-1</ncpoly> <ncpoly>x*x-1</ncpoly> <ncpoly>x*(y^5)*x*(y^2)*x*(y^4)*x*(y^3)-1</ncpoly> </basis> <Comment>The partial LLex Gb has 4 elements</Comment> <Comment>Other_groups_13a5b2c4d3</Comment> </FREEALGEBRA>