Difference between revisions of "ApCoCoA-1:Tetraeder group"
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(New page: === <div id="Tetraeder group">Tetraeder group</div> === ==== Description ==== The Tetraeder group is isomorphic to the alternating group A_4. Geo...) |
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| − | === <div id="Tetraeder group">[[:ApCoCoA:Symbolic data#Tetraeder_group|Tetraeder | + | === <div id="Tetraeder group">[[:ApCoCoA:Symbolic data#Tetraeder_group|Tetraeder Group]]</div> === |
==== Description ==== | ==== Description ==== | ||
The Tetraeder group is isomorphic to the alternating group A_4. Geometrically this group is generated by the | The Tetraeder group is isomorphic to the alternating group A_4. Geometrically this group is generated by the | ||
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Relations:=CreateRelationsTetraeder(); | Relations:=CreateRelationsTetraeder(); | ||
Gb:=NC.GB(Relations); | 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> | ||
Latest revision as of 20:59, 22 April 2014
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>
