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 ...)
 
 
(7 intermediate revisions by 3 users not shown)
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=== <div id="Higman_groups">[[:ApCoCoA:Symbolic data#Higman_group|Higman group]]</div> ===
+
=== <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>
(Reference: Edjvet, Rosenberger, Stille, Thomas, "On certain generalized tetrahedon groups", Computational And Geometric Aspects Of
+
 
Modern Algebra)
+
==== Reference ====
 +
Edjvet, Rosenberger, Stille, Thomas, "On certain generalized tetrahedon groups", Computational And Geometric Aspects Of Modern Algebra.
  
 
==== Computation ====
 
==== Computation ====
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   MEMORY.F2:=3;
 
   MEMORY.F2:=3;
 
   MEMORY.F3:=3;
 
   MEMORY.F3:=3;
 
+
   
  // a invers to d and b invers to e and c invers to f
 
 
   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;
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   Relations:=CreateRelationsDicyclic();
 
   Relations:=CreateRelationsDicyclic();
 
   Relations;
 
   Relations;
 +
 
 
   If Size(Relations) > 0 Then
 
   If Size(Relations) > 0 Then
     GB:=NC.GB(Relations,31,1,100,1000);
+
     Gb:=NC.GB(Relations,31,1,100,1000);
     Size(GB);
+
     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>