Tetrahedral symmetry

The ( alternating group of degree 4 ) is a specific 12 -element group that is studied in the mathematical branch of group theory. It is closely related to the symmetric group, it is in the subgroup consisting of all even permutations. Geometric arises as the group of rotations of the regular tetrahedron to itself.

Geometric introduction

Considering the rotations that transform a regular tetrahedron into itself, we find 12 options:

• The identity,
• Three rotations of 180 ° about the axes passing through the centers of two opposite edges,
• Four rotations by 120 ° around altitudes of the tetrahedron,
• Four rotations by 240 ° around altitudes of the tetrahedron.

Reflections are not considered here. For the rotations we choose the following names:

• Is the rotation of 180 ° around the straight line 12 and 34 passing through the centers of the edges ( 1,2,3 and 4 of tetrahedral corners as in the first drawing ).
• Is the rotation of 180 ° around the straight line 13 and 24 passing through the centers of the edges.
• Is the rotation of 180 ° around the straight line 14 and 23 passing through the centers of the edges.
• Is seen to extend through the corner height, in the positive direction (i.e., counterclockwise) of the pierced edge of the rotation by 120 °.
• Is the rotation by 240 ° around the line passing through the corner height, also with the direction of rotation indicated above.

These rotations can be combined by sequential execution to give back on a rotation from the list above. You just write two rotations (often without connectives, or with or ) next to each other, referring to that first the right-wing and then execute the left- rotation. The notation makes it clear that the rotation through 240 ° is equal to twice the sequential execution of the rotation by 120 °.

Is obtained in this way, the 12 -element group of all rotations of the regular tetrahedron to itself.

Plotting all the links so formed into a truth table a, we obtain

The permutation as

The rotations described above are already thereby determine how that are mapped to each other with 1,2,3 and 4 designated corners. Each element can therefore be considered as a permutation of the set. If one uses the standard two line shape and the Zykelschreibweise, we obtain:

One can see at a glance that each element of as a product of an even number of transpositions ( = Zweierpermutationen ) can be written. The corresponding permutations are called also straight, ie consists exactly of the even permutations of the set. This occurs as the core of the Signum Picture: on, the symmetric group of degree.

Properties

Subgroups

All subgroups are shown in the adjacent drawing.

Is isomorphic to the Klein four-group. According to the set of Lagrange divides the order of each subgroup the group order, in this case 12 Conversely, it does not however give any divisor of the group order is a subgroup of this order. This is an example of this phenomenon, because it has no subgroup of order 6

Normal subgroup, solvability

That is not abelian, but dissolvable, as the series

Shows. The sign means " is a normal subgroup in ".

Is the commutator subgroup of, in particular, therefore a normal subgroup and it is

Semi direct product

Since and are relatively prime group orders, it follows from the theorem of Schur - Zassenhaus that is isomorphic to the semi- direct product, the residue class maps to the automorphism.

Generators and relations

You may also be groups describe that a system of generators and relations, the growers must meet one indicates. Generators and relations one notes, by the | character separated in angle brackets. The group is then the free group generated by the generators modulo the normal subgroup generated by the relations. In this sense is:

It is easy to fulfill and that the relations and that the whole group and produce, but this is not yet sufficient for the proof.