Difference between revisions of "ApCoCoA-1:Oktaeder group"

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(New page: === <div id="Oktaeder group">Oktaeder group</div> === ==== Description ==== The Oktaeder group is a subgroup of the symmetric group. Like the Tetr...)
 
 
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=== <div id="Oktaeder group">[[:ApCoCoA:Symbolic data#Oktaeder_group|Oktaeder group]]</div> ===
+
=== <div id="Oktaeder group">[[:ApCoCoA:Symbolic data#Oktaeder_group|Oktaeder Group]]</div> ===
 
==== Description ====
 
==== Description ====
 
The Oktaeder group is a subgroup of the symmetric group. Like the Tetraeder group this group is generated
 
The Oktaeder group is a subgroup of the symmetric group. Like the Tetraeder group this group is generated
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   Relations:=CreateRelationsOktaeder();
 
   Relations:=CreateRelationsOktaeder();
 
   Gb:=NC.GB(Relations);
 
   Gb:=NC.GB(Relations);
 +
====Example in Symbolic Data Format====
 +
  <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 +
  <vars>a,b</vars>
 +
  <basis>
 +
  <ncpoly>a*a-1</ncpoly>
 +
  <ncpoly>b*b*b-1</ncpoly>
 +
  <ncpoly>(a*b)^4-1</ncpoly>
 +
  </basis>
 +
  <Comment>Oktaeder_group</Comment>
 +
  </FREEALGEBRA>

Latest revision as of 20:59, 22 April 2014

Description

The Oktaeder group is a subgroup of the symmetric group. Like the Tetraeder group this group is generated only by rotations.

 O = <a,b | a^2 = b^3 = (ab)^4 = 1>

Reference

Geometries and Transformations, Manuscript, Chapter 11: Finite symmetry groups, N.W. Johnson, 2011

Computation

 /*Use the ApCoCoA package ncpoly.*/
 
 Use ZZ/(2)[a,b];
 NC.SetOrdering("LLEX");

 Define CreateRelationsOktaeder()
   Relations:=[];
   // add the relation a^2 = 1 
   Append(Relations,[[a^2],[1]]);
   
   // add the relation b^3 = 1
   Append(Relations,[[b^3],[1]]);
   
   // add the relation (ab)^4 = 1
   Append(Relations,[[a,b,a,b,a,b,a,b],[1]]);
   
    Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsOktaeder();
 Gb:=NC.GB(Relations);

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier">
 	<vars>a,b</vars>
 	<basis>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>b*b*b-1</ncpoly>
 	<ncpoly>(a*b)^4-1</ncpoly>
 	</basis>
 	<Comment>Oktaeder_group</Comment>
 </FREEALGEBRA>