A4 (group) - A4 (Gruppe)

The ( alternating group of 4th degree) is a specific 12-element group that is examined in the mathematical sub-area of group theory . It is closely related to the symmetrical group , it is the around the subgroup that consists of all even permutations . Geometrically, theas a group of rotations of the regular tetrahedron on itself.

Geometrical introduction

The twists and of the Tetraeders

If one looks at the rotations that transform a regular tetrahedron into itself, one finds 12 possibilities: [1]

  • the identity ,
  • three rotations of 180 ° around axes that run through the centers of two opposite edges,
  • four rotations of 120 ° around the heights of the tetrahedron,
  • four rotations of 240 ° around the heights of the tetrahedron.

Reflections are not considered here. We choose the following terms for the rotations:

  • is the rotation by 180 ° around the straight line that runs through the midpoints of the edges 12 and 34 (1, 2, 3 and 4 denote tetrahedron corners as in the adjacent drawing).
  • is the rotation by 180 ° around the straight line that runs through the midpoints of edges 13 and 24.
  • is the rotation through 180 ° around the straight line that runs through the midpoints of the edges 14 and 23.
  • be the rotation of 120 ° around the corner running height, in the positive direction of rotation ( i.e. counterclockwise ) seen from the pierced corner.
  • be the rotation by 240 ° around the corner running height, also with the direction of rotation specified above.

These rotations can be combined by executing them one after the other , which again results in a rotation from the list above. You just write two rotations (often without a link, or with or ) next to each other and means that first the right-hand and then the left-hand rotation must be performed. The spelling Already makes clear that the rotation by 240 ° is equal to the double execution of the rotation by 120 °.

In this way the 12-element group is obtained of all rotations of the regular tetrahedron.

If you enter all the links formed in this way in a link table , you get


Linking table of the alternating group A 4 in color. The neutral element is black

The graphic on the right shows the link table in color. Such graphics show some relationships better than is the case with the use of numbers, letters or symbols. It should be noted that, in general, no particular arrangement can be identified for the elements of a group. A fixed rule, however, is that the neutral element is the first element of every row and column (top left corner). This colored link table follows the order of the elements in the table on the left. Colored link tables as shown in the graphic are used in the MathWorld online encyclopedia for mathematics , as are those in grayscale. [2]

Representation as a permutation group

The rotations described above are already determined by how the corners marked 1, 2, 3 and 4 are mapped onto one another. Every element of the can therefore be used as a permutation of the set be understood. If you use the usual two-line form and the cycle notation , you get:

You can see at a glance that every element of the can be written as a product of an even number of transpositions (= two permutations). The associated permutations are also called straight , that is, the consists exactly of the even permutations of the set . With that theas the core of the Signum image: on, where is the fourth degree symmetric group .

characteristics

Subgroups

The subgroups of the

All subgroups of the [3] are given in the adjacent drawing.

is isomorphic to Klein's group of four . According to Lagrange's theorem , the order of each subgroup divides the group order , in this case 12. Conversely, however, there need not be a subgroup of this order for every divisor of the group order. The is an example of this phenomenon because it has no subgroup of order 6.

Normal divisor, solvability

The is not Abelian , because

but is resolvable , like the series

shows. The signmeans “ is normal divisor in” .

is the commutator group of, [4] especially a normal divisor and it holds

The two- and three-element subgroups are not normal divisors.

Semi-direct product

Gives and have coprime group orders, it follows from the Schur-Zassenhaus theorem that theto the semi-direct product is isomorphic, where the rest of the class on automorphism maps.

Generators and relations

Groups can also be described by specifying a generating system and relations that the generators must fulfill. Generators and relations are noted with the sign | separated, in angle brackets. The group is then the free group generated by the generators modulo the normal divisor generated by the relations. In this sense: [5]

It's easy to see that and fulfill the relations and that and generate the entire group, which is not sufficient for the proof.

Charactertafel

The character board of thelooks like this: [6]

See also

Individual evidence

  1. Arno Mitschka: Elements of group theory , study books mathematics (1975), ISBN 3-451-16528-7 , section X, solution to IV.7
  2. MathWorld: Tetrahedral Group On this website there are the link panels (in color) of the tetrahedral group and that of its subgroup, the tetrahedral rotating group that are isomorphic to the alternating group is. The order of the elements for the color graphics is not specified there.
  3. ^ PJ Pahl, R. Damrath: Mathematical foundations of engineering informatics , Springer-Verlag (2000), ISBN 3-540-60501-0 , section 7.8.3. example 1
  4. ^ K. Meyberg: Algebra, Part I , Carl Hanser Verlag (1980), ISBN 3-446-13079-9 , example 2.6.4
  5. K. Lamotke: Regular Solids and Isolated Singularities, Vieweg-Verlag (1986), ISBN 3-528-08958-X, Kapitel I §8: Generators and Relations for the Finite Subgroups of SO(3)
  6. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag (1976), ISBN 3-446-12172-2 , example 9.7.1 c