ApCoCoA:Tetraeder group

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Description

The Tetraeder group is isomorphic to the alternating group A_4. Geometrically this group is generated by the rotations of a regular tetrahedron. Its representation is given by

 A_4 = <a,b | a^2 = b^3 = (ab)^3 = 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 CreateRelationsTetraeder()
   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)^3 = 1
   Append(Relations,[[a,b,a,b,a,b],[1]]);
   
    Return Relations;
 EndDefine;
 
 Relations:=CreateRelationsTetraeder();
 Gb:=NC.GB(Relations);

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-01-24" createdBy="strohmeier">
 	<vars>a,b</vars>
 	<basis>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>b*b*b-1</ncpoly>
 	<ncpoly>(a*b)^3-1</ncpoly>
 	</basis>
 	<Comment>Tetraeder_group</Comment>
 </FREEALGEBRA>