Dihedral group of order 6

Referred to in the mathematical branch of group theory a certain symmetric group with 6 elements. Alternative names are, or ( dihedral group ). Geometrically defined as group of Kongruenzabbildungen of the equilateral triangle formed up.

Introduction

Looking at the Kongruenzabbildungen that transform an equilateral triangle in itself, one finds 6 options:

  • The identity mapping,
  • The rotation through 120 ° about the center of the triangle
  • The rotation by 240 ° around the center of the triangle
  • Three reflections and of the three perpendicular bisectors of the triangle.

This Kongruenzabbildungen can be combined by sequential execution, yielding once again receives a congruence. You just write two Kongruenzabbildungen (often without connectives, or with or ) next to each other, referring to that

Congruence is to be executed. 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 group of all sechselementige Kongruenzabbildungen of the equilateral triangle up. Plotting all the links so formed into a truth table a, we obtain

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

Generalizing this construction by the equilateral triangle is replaced by a regular -gon, one comes to the concept of dihedral. Therefore, the group discussed here is also referred to.

Elements of the permutations

A congruence of the equilateral triangle is thus already clearly defined, like the corners labeled 1, 2 and 3 are mapped to each other. Each element can therefore be considered as a permutation of the set. You can see below, the first two lines form and behind the Zykelschreibweise of the elements and their orders:

Properties

No abelian group

The group is not abelian group as above truth table can be removed; example applies. It is up to isomorphism the smallest non - abelian group, that is, every non- abelian group is isomorphic to either or has more elements.

Subgroups and normal subgroups

The sub-groups in addition to the trivial subgroups and are themselves:

  • . This sub-group is a normal divider and is also referred to as alternating group grade 3.
  • . These subgroups are not normal subgroups; for example.

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:

Irreducible representations

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

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

Although you get a different two-dimensional representation, if you replaced, but this is equivalent to the specified.

Other examples

General linear group over Z / 2

The general linear group of degree 2 over the residue field,

Is isomorphic to.

Transformation group

The fractional linear transformations with coefficients in an arbitrary field and the assignments

Produce with the sequential execution as a group linking a group that is isomorphic to. The remaining four members of the group are:

The truth table is as above. The six group members differ in a setting of elements

In the tables of values ​​, if has at least 5 elements.

Quotient group Group representation Parity of a permutation List of small groups
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