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 Chevalley’s theorem regarding the invariants of a finite reflection group.

Title	dihedral group
Canonical name	DihedralGroup
Date of creation	2013-03-22 12:22:53
Last modified on	2013-03-22 12:22:53
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	15
Author	rmilson (146)
Entry type	Definition
Classification	msc 20F55
Related topic	Symmetry2