S3 (group)

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In the mathematical sub-area of group theory, the symmetrical group denotes a specific group with 6 elements. It can be described as a group of the six permutations of a three-element set. Alternative names are and . It is isomorphic with the Dïedergruppe  , the group of congruence maps of the equilateral triangle on itself.


The effects of the pictures , , , and

If you look at the congruence maps that transform an equilateral triangle into themselves, you find 6 possibilities:

  • the identical figure ,
  • the rotation of 120 ° around the center of the triangle,
  • the rotation of 240 ° around the center of the triangle,
  • the three reflections and at the three perpendiculars of the triangle.

These congruence maps can be combined by executing them one after the other , whereby a congruence map is obtained again. One simply writes two congruence maps (often without linking symbols, or with or ) next to each other and means that

first the one on the right and then the one on the left

Congruence mapping is to be carried out. The notation makes it clear that the rotation by 240 ° is equal to the double execution of the rotation by 120 °.

In this way the alternate-element group of all congruence maps of the equilateral triangle is obtained. If you enter all the links formed in this way in a link table , you get

Linking table of the symmetrical group S 3 in color. The neutral element (the identical picture) is black

If you want to the product of two elements of figure out, so they were looking in the truth table first with marked column, with marked line on; at the intersection of this column and row is the product.

The graphic on the right shows the link table in color. This colored link table follows the order of the elements in the table on the left. Colored link tables, as in the graphic, are used in the online encyclopedia for mathematics MathWorld , as are those in grayscale.

If one generalizes this construction by replacing the equilateral triangle with a regular corner, one arrives at the concept of the dihedral group . Therefore, the group discussed here is also referred to as.

Elements of the S 3 as permutations

A congruence mapping of the equilateral triangle is already clearly defined by how the corners labeled 1, 2 and 3 are mapped onto one another. Each element of can therefore be understood as a permutation of the set . In the following, the two-line form is given first, followed by the cycle notation of the elements and their order :


Not an Abelian group

The group is not an Abelian group , as can be seen from the table above (it is not symmetrical to the main diagonal); for example . Except for isomorphism, it is the smallest non-Abelian group, that is, every non-Abelian group is either isomorphic to or has more elements.

Subgroups and normal divisors

The subgroups next to the trivial subgroups and themselves are:

  • . This subgroup (the group of rotations) is a normal divisor and is also known as an alternating group of degree 3.
  • . These subgroups (the groups of reflections) are not normal subgroups; for example is .
  • The center of is trivial (consists only of ). Thus, an element that is different from one another commutes only with powers of itself.

Generators and Relations

Groups can also be described by specifying a system of generators and relations that the generators must fulfill. Generators and relations are noted, separated by the symbol     , 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:

Irreducible representations

Except for equivalence, it has three irreducible representations , two one-dimensional and one two-dimensional. To specify these representations, it is sufficient to specify the images of and , because these elements create the group.

  • The trivial representation:
  • The Signum illustration:
  • The two-dimensional representation: .

While you get a different two-dimensional representation when by replaced, but this is equivalent to the specified. These considerations lead to the following character table :

Further examples

General linear group over

The general linear group 2 of degree over the residue field , is isomorphic to .

Transformation group

The fractional linear functions with coefficients from any field and the assignments

create a group that is isomorphic to the sequential execution as a group link . The other 4 group members are:

The link panel is as above .
The 6 group members differ in the use of elements

also in the value tables if has at least 5 elements.

Automorphism group

This is isomorphic to the automorphism group of Klein's group of four . This follows easily from the observation that every permutation of the three elements of order 2 of the Klein group of four defines an automorphism.

See also

Web links

Individual evidence

  1. Arno Mitschka: Elements of group theory. Study books mathematics, 1975, ISBN 3-451-16528-7 , section II.5
  2. a b This order comes from the operator perspective, as it prevails with the sequential connection of mappings (also with the permutations). The sequence is irrelevant for pure group theory.
  3. MathWorld: Symmetric Group The symmetric group S 3 is not included, but S 4 for example.
  4. ^ K. Meyberg: Algebra, Part I. Carl Hanser Verlag, 1980, ISBN 3-446-13079-9 , example 2.4.2.c
  5. ^ K. Meyberg: Algebra, Part I. Carl Hanser Verlag, 1980, ISBN 3-446-13079-9 , Example 2.7.18.c
  6. JP Serre: Representations of finite groups. Vieweg, 1972, ISBN 3-528-03556-0 , §5.3
  7. ^ Kurt Meyberg: Algebra II. Carl Hanser Verlag, 1976, ISBN 3-446-12172-2 , Example 9.7.1 b
  8. If the body of the complex numbers, more precisely: the Riemann number ball , then it is a Möbius transformation .