Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order. In abstract algebra, refers to this same dihedral group. This article uses the geometric convention,.

Definition

The word "dihedral" comes from "di-" and "-hedron".
The latter comes from the Greek word hédra, which means "face of a geometrical solid". Overall, it thus refers to the two faces of a polygon.

Elements

A regular polygon with sides has different symmetries: rotational symmetries and reflection symmetries; here,. The associated rotations and reflections make up the dihedral group. If is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If is even, there are axes of symmetry connecting the midpoints of opposite sides and axes of symmetry connecting opposite vertices. In either case, there are axes of symmetry and elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.

Group structure

As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group.
The following Cayley table shows the effect of composition in the dihedral group of order 6, —the symmetries of an equilateral triangle. Here, denotes the identity, and denote counterclockwise rotations by 120° and 240° respectively, as well as,, and denote reflections across the three lines shown in the adjacent picture.

	r₀	r₁	r₂	s₀	s₁	s₂
r₀	r₀	r₁	r₂	s₀	s₁	s₂
r₁	r₁	r₂	r₀	s₁	s₂	s₀
r₂	r₂	r₀	r₁	s₂	s₀	s₁
s₀	s₀	s₂	s₁	r₀	r₂	r₁
s₁	s₁	s₀	s₂	r₁	r₀	r₂
s₂	s₂	s₁	s₀	r₂	r₁	r₀

For example,, because the reflection followed by the reflection s₂ results in a rotation of 120°. The order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative.
In general, the group has elements and, with composition given by the following formulae:
In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus.

Matrix representation

Centering the regular polygon at the origin, elements of the dihedral group act as linear transformations of the plane. This lets elements of represented as matrices, with composition being matrix multiplication. This is an example of a group representation.
For example, the elements of the group dihedral group of order 8, —the group symmetry of a square—can be represented by the following eight matrices:
Here, these matrices represents the symmetries of an axis-aligned square centered at the origin, which acts on the plane by multiplication on column vectors of coordinates. The element represents the identity. The elements and represents the reflection across horizontal and vertical axis. The elements and represents the reflection across diagonals. Three other elements,, and are rotations around a center.
In general, the matrices for elements of have the following form:
Here, the element is a rotation matrix, expressing a counterclockwise rotation through an angle of. The element is a reflection across a line that makes an angle of with the.

Other definitions

is the semidirect product of acting on via the automorphism.
It hence has presentation
Using the relation, we obtain the relation.
It follows that is generated by and. This substitution also shows that has the presentation
In particular, belongs to the class of Coxeter groups.

Small dihedral groups

is isomorphic to, the cyclic group of order 2.
is isomorphic to, the Klein four-group.
and are exceptional in that:

and are the only abelian dihedral groups. Otherwise, is non-abelian.
is a subgroup of the symmetric group for. Since for or, for these values, is too large to be a subgroup.
The inner automorphism group of is trivial, whereas for other even values of, this is.

The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.

D₁ = Z₂	D₂ = Z₂² = K₄	D₃	D₄	D₅


D D × Z	D	D₈	D₉	D D × Z

D₃ = S₃	D₄

The dihedral group as symmetry group in 2D and rotation group in 3D

An example of abstract group, and a common way to visualize it, is the group of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. consists of rotations of multiples of about the origin, and reflections across lines through the origin, making angles of multiples of with each other. This is the symmetry group of a regular polygon with sides.
is generated by a rotation of order and a reflection of order 2 such that
In geometric terms: in the mirror a rotation looks like an inverse rotation.
In terms of complex numbers: multiplication by and complex conjugation.
In matrix form, by setting
and defining and for we can write the product rules for D_n as
The dihedral group D₂ is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. The elements of D₂ can then be represented as, where e is the identity or null transformation and rs is the reflection across the y-axis.
D₂ is isomorphic to the Klein four-group.
For n > 2 the operations of rotation and reflection in general do not commute and D_n is not abelian; for example, in D₄, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.
Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.
The elements of can be written as,,,...,,,,,..., . The first listed elements are rotations and the remaining elements are axis-reflections. The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection.
So far, we have considered to be a subgroup of, i.e. the group of rotations and reflections of the plane. However, notation is also used for a subgroup of SO which is also of abstract group type : the proper symmetry group of a regular polygon embedded in three-dimensional space. Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore, it is also called a dihedron, which explains the name dihedral group.

Examples of 2D dihedral symmetry

Properties

The properties of the dihedral groups with depend on whether is even or odd. For example, the center of consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and the element r^n/2.
In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.
For n twice an odd number, the abstract group is isomorphic with the direct product of and.
Generally, if m divides n, then has n/''m subgroups of type, and one subgroup m''. Therefore, the total number of subgroups of , is equal to d + σ, where d is the number of positive divisors of n and σ is the sum of the positive divisors of n. See list of small groups for the cases n ≤ 8.
The dihedral group of order 8 is the smallest example of a group that is not a T-group. Any of its two Klein four-group subgroups has as normal subgroup order-2 subgroups generated by a reflection in D₄, but these subgroups are not normal in D₄.