Difference between revisions of "ApCoCoA-1:OrdinaryTetrahedron groups"
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(New page: === <div id="Higman_groups">Higman group</div> === ==== Description ==== An Ordinary Tetrahedron group is a group with a representation of the form ...) |
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| − | === <div id=" | + | === <div id="Ordinary_tetrahedron_groups">[[:ApCoCoA:Symbolic data#Ordinary_tetrahedron_groups|Ordinary Tetrahedron Groups]]</div> === |
==== Description ==== | ==== Description ==== | ||
An Ordinary Tetrahedron group is a group with a representation of the form | An Ordinary Tetrahedron group is a group with a representation of the form | ||
G(e_1,e_2,e_3,f_1,f_2,f_3) = <x,y,z | x^{e_1} = y^{e_2} = z^{e_3} = (xy^{-1})^{f_1} = (yz^{-1})^{f_2} = (zx^{-1})^{f_3} = 1> | G(e_1,e_2,e_3,f_1,f_2,f_3) = <x,y,z | x^{e_1} = y^{e_2} = z^{e_3} = (xy^{-1})^{f_1} = (yz^{-1})^{f_2} = (zx^{-1})^{f_3} = 1> | ||
| − | + | ||
| − | Modern Algebra | + | ==== Reference ==== |
| + | Edjvet, Rosenberger, Stille, Thomas, "On certain generalized tetrahedon groups", Computational And Geometric Aspects Of Modern Algebra. | ||
==== Computation ==== | ==== Computation ==== | ||
| Line 16: | Line 17: | ||
MEMORY.F2:=3; | MEMORY.F2:=3; | ||
MEMORY.F3:=3; | MEMORY.F3:=3; | ||
| − | + | ||
| − | |||
Use ZZ/(2)[x,y,z]; | Use ZZ/(2)[x,y,z]; | ||
NC.SetOrdering("LLEX"); | NC.SetOrdering("LLEX"); | ||
| + | |||
Define CreateRelationsDicyclic() | Define CreateRelationsDicyclic() | ||
Relations:=[]; | Relations:=[]; | ||
| Line 38: | Line 39: | ||
EndFor; | EndFor; | ||
Append(Relations,[RelationA,[1]]); | Append(Relations,[RelationA,[1]]); | ||
| + | |||
// add the relation (yz^{-1})^{f_2} | // add the relation (yz^{-1})^{f_2} | ||
RelationB:=[]; | RelationB:=[]; | ||
| Line 53: | Line 55: | ||
EndFor; | EndFor; | ||
Append(Relations,[RelationC,[1]]); | Append(Relations,[RelationC,[1]]); | ||
| − | EndIf; | + | EndIf; |
| + | |||
Return Relations; | Return Relations; | ||
EndDefine; | EndDefine; | ||
| Line 59: | Line 62: | ||
Relations:=CreateRelationsDicyclic(); | Relations:=CreateRelationsDicyclic(); | ||
Relations; | Relations; | ||
| + | |||
If Size(Relations) > 0 Then | If Size(Relations) > 0 Then | ||
| − | + | Gb:=NC.GB(Relations,31,1,100,1000); | |
| − | Size( | + | Size(Gb); |
EndIf; | EndIf; | ||
| + | |||
| + | ====Example in Symbolic Data Format==== | ||
| + | <FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> | ||
| + | <vars>x,y,z</vars> | ||
| + | <uptoDeg>15</uptoDeg> | ||
| + | <basis> | ||
| + | <ncpoly>x^3-1</ncpoly> | ||
| + | <ncpoly>y^3-1</ncpoly> | ||
| + | <ncpoly>z^3-1</ncpoly> | ||
| + | <ncpoly>((x*y^(3-1))^(3))-1</ncpoly> | ||
| + | <ncpoly>((y*z^(3-1))^(3))-1</ncpoly> | ||
| + | <ncpoly>((z*x^(3-1))^(3))-1</ncpoly> | ||
| + | </basis> | ||
| + | <Comment>The partial LLex Gb has 140 elements</Comment> | ||
| + | <Comment>Ordinary_Tetrahedron_group_3</Comment> | ||
| + | </FREEALGEBRA> | ||
Latest revision as of 20:50, 22 April 2014
Description
An Ordinary Tetrahedron group is a group with a representation of the form
G(e_1,e_2,e_3,f_1,f_2,f_3) = <x,y,z | x^{e_1} = y^{e_2} = z^{e_3} = (xy^{-1})^{f_1} = (yz^{-1})^{f_2} = (zx^{-1})^{f_3} = 1>
Reference
Edjvet, Rosenberger, Stille, Thomas, "On certain generalized tetrahedon groups", Computational And Geometric Aspects Of Modern Algebra.
Computation
/*Use the ApCoCoA package ncpoly.*/
// Variables of Ordinary Tetrahedon group
MEMORY.E1:=3;
MEMORY.E2:=3;
MEMORY.E3:=3;
MEMORY.F1:=3;
MEMORY.F2:=3;
MEMORY.F3:=3;
Use ZZ/(2)[x,y,z];
NC.SetOrdering("LLEX");
Define CreateRelationsDicyclic()
Relations:=[];
If MEMORY.E1 < 2 Or MEMORY.E2 < 2 Or MEMORY.E3 < 2 Or MEMORY.F1 < 2 Or MEMORY.F2 < 2 Or MEMORY.F3 < 2 Then
Output:="Wrong Input! Please check that E_i and F_i are greater than 1";
Print(Output);
Else
// add the relations x^{e_1} = 1, y^{e_2} = 1 and z^{e_3} = 1
Append(Relations,[[x^(MEMORY.E1)],[1]]);
Append(Relations,[[y^(MEMORY.E2)],[1]]);
Append(Relations,[[z^(MEMORY.E3)],[1]]);
// add the relation (xy^{-1})^{f_1}
RelationA:=[];
For Index1 := 1 To MEMORY.F1 Do
Append(RelationA,x);
Append(RelationA,y^(MEMORY.E2-1));
EndFor;
Append(Relations,[RelationA,[1]]);
// add the relation (yz^{-1})^{f_2}
RelationB:=[];
For Index2 := 1 To MEMORY.F2 Do
Append(RelationB,y);
Append(RelationB,z^(MEMORY.E3-1));
EndFor;
Append(Relations,[RelationB,[1]]);
// add the relation (zx^{-1})^{f_3}
RelationC:=[];
For Index3 := 1 To MEMORY.F3 Do
Append(RelationC,z);
Append(RelationC,x^(MEMORY.E1-1));
EndFor;
Append(Relations,[RelationC,[1]]);
EndIf;
Return Relations;
EndDefine;
Relations:=CreateRelationsDicyclic();
Relations;
If Size(Relations) > 0 Then
Gb:=NC.GB(Relations,31,1,100,1000);
Size(Gb);
EndIf;
Example in Symbolic Data Format
<FREEALGEBRA createdAt="2014-03-02" createdBy="strohmeier"> <vars>x,y,z</vars> <uptoDeg>15</uptoDeg> <basis> <ncpoly>x^3-1</ncpoly> <ncpoly>y^3-1</ncpoly> <ncpoly>z^3-1</ncpoly> <ncpoly>((x*y^(3-1))^(3))-1</ncpoly> <ncpoly>((y*z^(3-1))^(3))-1</ncpoly> <ncpoly>((z*x^(3-1))^(3))-1</ncpoly> </basis> <Comment>The partial LLex Gb has 140 elements</Comment> <Comment>Ordinary_Tetrahedron_group_3</Comment> </FREEALGEBRA>
