Difference between revisions of "ApCoCoA-1:Other13 groups"
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(New page: === <div id="Other13_groups">Other groups</div> === ==== Description ==== This group has the following finite representation: G = <x,y | x^2 = xy^{...) |
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| − | === <div id="Other13_groups">[[:ApCoCoA:Symbolic data#Other_groups|Other | + | === <div id="Other13_groups">[[:ApCoCoA:Symbolic data#Other_groups|Other Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
This group has the following finite representation: | This group has the following finite representation: | ||
| Line 11: | Line 11: | ||
/*Use the ApCoCoA package ncpoly.*/ | /*Use the ApCoCoA package ncpoly.*/ | ||
| − | // Note that a,b >= 1 | + | // Note that a,b,c,d >= 1 |
MEMORY.A := 3; | MEMORY.A := 3; | ||
MEMORY.B := 3; | MEMORY.B := 3; | ||
MEMORY.C := 4; | MEMORY.C := 4; | ||
MEMORY.D := 5; | MEMORY.D := 5; | ||
| + | |||
// y is invers to z, the invers element of x follows directly from the relation x^2 = 1 | // y is invers to z, the invers element of x follows directly from the relation x^2 = 1 | ||
Use ZZ/(2)[x,y,z]; | Use ZZ/(2)[x,y,z]; | ||
| Line 31: | Line 32: | ||
// add the relation xy^{a}xy^{b}xy^{c}xy^{d} | // 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]]); | Append(Relations,[[x,y^(MEMORY.A),x,y^(MEMORY.B),x,y^(MEMORY.C),x,y^(MEMORY.D)],[1]]); | ||
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
Relations:=CreateRelationsOther13(); | 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> | ||
Latest revision as of 21:10, 22 April 2014
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>
