Difference between revisions of "ApCoCoA-1:Other12 groups"
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(New page: === <div id="Other12_groups">Other groups</div> === ==== Description ==== This group has the following finite representation: G = <x,t | tx^{a}t^{-...) |
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| − | === <div id="Other12_groups">[[:ApCoCoA:Symbolic data#Other_groups|Other | + | === <div id="Other12_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 16: | Line 16: | ||
MEMORY.B := 3; | MEMORY.B := 3; | ||
MEMORY.N := 4; | MEMORY.N := 4; | ||
| + | |||
// x is invers to z, t has an implicit invers (Relation: t^{n} = 1) | // x is invers to z, t has an implicit invers (Relation: t^{n} = 1) | ||
Use ZZ/(2)[x,t,z]; | Use ZZ/(2)[x,t,z]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsOther12() | Define CreateRelationsOther12() | ||
Relations:=[]; | Relations:=[]; | ||
| Line 39: | Line 41: | ||
Append(RelationBuffer1,t^(MEMORY.N-1)); | Append(RelationBuffer1,t^(MEMORY.N-1)); | ||
Append(Relations,[RelationBuffer1,[x^MEMORY.B]]); | Append(Relations,[RelationBuffer1,[x^MEMORY.B]]); | ||
| − | + | ||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
Relations:=CreateRelationsOther12(); | 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> | ||
Latest revision as of 21:10, 22 April 2014
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>
