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From Wikipedia
In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both 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.
See also: Dihedral symmetry in three dimensions.
Notation
There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted D_{n}, while in algebra the same group is denoted by D_{2n} to indicate the number of elements.
In this article, D_{n} (and sometimes Dih_{n}) refers to the symmetries of a regular polygon with n sides.
Definition
Elements
A regular polygon with n sides has 2n different symmetries: nrotational symmetries and nreflection symmetries. The associated rotations and reflections make up the dihedral group D_{n}. If n is odd each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and 2n 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. The following picture shows the effect of the sixteen elements of D_{8} on a stop sign:
The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections.
Group structure
As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry. This operation gives the symmetries of a polygon the algebraic structure of a finite group.
The following Cayley table shows the effect of composition in the group D_{3} (the symmetries of an equilateral triangle). R_{0} denotes the identity; R_{1} and R_{2} denote counterclockwise rotations by 120 and 240 degrees; and S_{0}, S_{1}, and S_{2} denote reflections across the three lines shown in the picture to the right.
For example, S_{2}S_{1} = R_{1} because the reflection S_{1} followed by the reflection S_{2} results in a 120degree rotation. (This is the normal backwards order for composition.) Note that the composition operation is not commutative.
In general, the group D_{n} has elements R_{0},...,R_{n−1} and S_{0},...,S_{n−1}, with composition given by the following formulae:
 R_i\,R_j = R_{i+j},\;\;\;\;R_i\,S_j = S_{i+j},\;\;\;\;S_i\,R_j = S_{ij},\;\;\;\;S_i\,S_j = R_{ij}.
In all cases, addition and subtraction of subscripts should be performed using modular arithmetic with modulus n.
Matrix representation
If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane. This lets us represent elements of D_{n} as matrices, with composition being matrix multiplication. This is an example of a (2dimensional) group representation.
For example, the elements of the group D_{4} can be represented by the following eight matrices:
 \begin{matrix}
R_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & R_1=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr), & R_2=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & R_3=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr), \\[1em] S_0=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & S_1=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr), & S_2=\bigl(\begin{smallmatrix}1&0\\[0.2em]0&1\end{smallmatrix}\bigr), & S_3=\bigl(\begin{smallmatrix}0&1\\[0.2em]1&0\end{smallmatrix}\bigr). \end{matrix}
In general, the matrices for elements of D_{n} have the following form:
 \begin{align}
R_k & = \begin{pmatrix} \cos \frac{2\pi k}{n} & \sin \frac{2\pi k}{n} \\ \sin \frac{2\pi k}{n} & \cos \frac{2\pi k}{n} \end{pmatrix} \ \ \text{and} \\ S_k & = \begin{pmatrix} \cos \frac{2\pi k}{n} & \sin \frac{2\pi k}{n} \\ \sin \frac{2\pi k}{n} & \cos \frac{2\pi k}{n} \end{pmatrix} . \end{align}
R_{k}is arotation matrix, expressing a counterclockwise rotation through an angle of . S_{k}is a reflection across a line that makes an angle of with the xaxis.
Small dihedral groups
For n = 1 we have Dih_{1}. This notation is rarely used except in the framework of the series, because it is equal to Z_{2}. For n = 2 we have Dih_{2}, the Klein fourgroup. Both are exceptional within the series: