Dihedral group

In group theory, the dihedral group is the isometry group of a regular polygon in the plane. The group contains elements, namely rotations and reflections. It is for non- Abelian. The name derives from the word dihedral ( hyphenation: dihedral, pronunciation [ the dər ː ] ) (Greek: Zweiflächner ) for regular - corner from.

Dihedral groups occur frequently in geometry and group theory. They are generated by two reflections ( elements of order ) and are thus the simplest examples of Coxeter groups.

Designations

There are two different names for dihedral groups. In geometry one usually writes about the connection with the regular -gon to emphasize. In group theory to write often to instead highlight the number of elements. This ambiguity, however, can easily fix by an explanatory supplement. In this article stands for the dihedral group with elements.

Definition

The dihedral group is the isometry group of a regular -gon in the plane. This consists of rotations and reflections, so it has a total of elements. The isometries are also known as symmetry transformations. The linkage of the group is given by the sequential execution of symmetry transformations.

Examples

An example is the dihedral group of Kongruenzabbildungen an equilateral triangle on itself, which is also called symmetric group. is in accordance with the symmetry group of the square under reflections and rotations.

Is isomorphic to the Klein four-group and the symmetry group ( consisting only of reflections and the identity) of four points of a square, in which only the right and left side are drawn (ie two two corner). is the symmetry group of a biangle.

The following graphic illustrates the dihedral group on the basis of rotations and reflections of a stop sign: The first row shows all eight rotations, the second line all eight reflections.

Matrix representation

We consider a planar regular -gon. We choose its center as the origin of a coordinate system, any of its axes of symmetry as - axis and the normal to it ( in the usual orientation, so that a legal system yields ) as axis. The dihedral group can then be easily represented as a matrix group. Furthermore, according to the rotation of the angle and the reflection in the line through which is inclined at an angle relative to the positive axis. As matrices, these transformations then write this:

• And. Therefore, we can restrict ourselves to.
• The rotation through the angle, is the identity.
• Is the rotation through the angle and it applies to all.
• Is the reflection in the -axis, and it applies to all.

If it is odd, then each of the mirror axis extending through a vertex and the midpoint of the opposite side. For straight however, there are two types of mirror axes by two opposite corners, or by two opposing midpoints.

In this illustration, writing, for example, the eight elements of the dihedral group as follows:

These rotations and reflections can be pictorially represented as follows:

Permutation representation

Let us first consider the example of the dihedral group. This operates shown by symmetry transformations to a square as in the previous graph. If we consider the action of the dihedral group on the vertices, we obtain a faithful representation of the symmetric group, ie an injective group homomorphism. Specifically, the transformations act on the corners as the following permutations:

More generally defines the operation of the dihedral group on the vertices of a faithful representation. In the above notation we obtain, for example, the permutation

In Zykelschreibweise this is the cyclic permutation which maps to, at, and so on until finally, is shown. The other rotations are obtained from this by means of the relation for all. For the reflections obtained in accordance Zykelschreibweise

With straight or

For odd. The other reflections are obtained from this by means of the relation for all.

Generators and relations

All rotations are generated by. These form a cyclic subgroup of order and therefore of index. The group as a whole is obtained by adding any reflection, for example. This gives the presentation

The concatenation of two reflections is a rotation; is the angle between the two Spieglungsachsen, its concatenation is a rotation by the angle. This means that the two adjacent dihedral reflections, for example, and is generated. This gives the presentation

This is the simplest case of a Coxeter group.

Applications

Geometry

Dihedral groups are the simplest examples of reflection groups. They play in the classical geometry play an important role, for example in the classification of regular polyhedra. In dimension here corresponds to the dihedral regular polygons.

Encoding

The defined by the above permutations number shortcut is used in checksum as an alternative to various modulo - based methods. For example, had the German banknotes dihedral checksums.