# Difference between revisions of "ApCoCoA-1:Tetraeder group"

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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 |

## 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>