# dihedral group

The $n^{\text{th}}$ dihedral group is the symmetry group of the regular $n$-sided polygon. The group consists of $n$ reflections, $n-1$ rotations, and the identity transformation. In this entry we will denote the group in question by $\mathcal{D}_{n}$. An alternate notation is $\mathcal{D}_{2n}$; in this approach, the subscript indicates the order of the group.

Letting $\omega=\exp(2\pi i/n)$ denote a primitive $n^{\text{th}}$ root of unity, and assuming the polygon is centered at the origin, the rotations $R_{k},\;k=0,\ldots,n-1$ (Note: $R_{0}$ denotes the identity) are given by

 $R_{k}:z\mapsto\omega^{k}z,\quad z\in\mathbb{C},$

and the reflections $M_{k},\;k=0,\ldots,n-1$ by

 $M_{k}:z\mapsto\omega^{k}\bar{z},\quad z\in\mathbb{C}$

The abstract group structure is given by

 $\displaystyle R_{k}R_{l}$ $\displaystyle=R_{k+l},$ $\displaystyle R_{k}M_{l}$ $\displaystyle=M_{k+l}$ $\displaystyle M_{k}M_{l}$ $\displaystyle=R_{k-l},$ $\displaystyle M_{k}R_{l}$ $\displaystyle=M_{k-l},$

where the addition and subtraction is carried out modulo $n$.

The group can also be described in terms of generators and relations as

 $\left(M_{0}\right)^{2}=\left(M_{1}\right)^{2}=(M_{1}M_{0})^{n}=\mathrm{id}.$

This means that $\mathcal{D}_{n}$ is a rank-1 Coxeter group.

Since the group acts by linear transformations

 $(x,y)\to(\hat{x},\hat{y}),\quad(x,y)\in\mathbb{R}^{2}$

there is a corresponding action on polynomials $p\to\hat{p}$, defined by

 $\hat{p}(\hat{x},\hat{y})=p(x,y),\quad p\in\mathbb{R}[x,y].$

The polynomials left invariant by all the group transformations form an algebra. This algebra is freely generated by the following two basic invariants:

 $x^{2}+y^{2},\quad x^{n}-\binom{n}{2}x^{n-2}y^{2}+\cdots,$

the latter polynomial being the real part of $(x+iy)^{n}$. It is easy to check that these two polynomials are invariant. The first polynomial describes the distance of a point from the origin, and this is unaltered by Euclidean reflections through the origin. The second polynomial is unaltered by a rotation through $2\pi/n$ radians, and is also invariant with respect to complex conjugation. These two transformations generate the $n^{\text{th}}$ dihedral group. Showing that these two invariants polynomially generate the full algebra of invariants is somewhat trickier, and is best done as an application of Chevalleyβs theorem regarding the invariants of a finite reflection group.

Title dihedral group DihedralGroup 2013-03-22 12:22:53 2013-03-22 12:22:53 rmilson (146) rmilson (146) 15 rmilson (146) Definition msc 20F55 Symmetry2