PlanetMath: dihedral group

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dihedral group

(Definition)

The $n^{\text{th}}$ dihedral group is the symmetry group of the regular -sided polygon. The group consists of reflections, 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: denotes the identity) are given by

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

and the reflections $M_k,\; k=0,\ldots,n-1$ by

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

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

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

$\displaystyle \left(M_0\right)^2 =\left(M_1\right)^2 = (M_1 M_0)^n = \mathrm{id}.$

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

Since the group acts by linear transformations

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

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

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

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

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

the latter polynomial being the real part of

. 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.

"dihedral group" is owned by rmilson. [ full author list (2) ]

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See Also: symmetry

Attachments:: octic group (Example) by Daume
dihedral group properties (Topic) by Algeboy

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Cross-references: finite, generate, complex conjugation, radians, Euclidean, point, distance, real part, freely generated, algebra, invariant, polynomials, action, linear transformations, Coxeter group, generators and relations, terms, subtraction, addition, structure, origin, root of unity, primitive, order, subscript, transformation, identity, rotations, reflections, polygon, group, symmetry
There are 24 references to this entry.

This is version 12 of dihedral group, born on 2002-02-18, modified 2007-06-12.
Object id is 2159, canonical name is DihedralGroup.
Accessed 8957 times total.

Classification:

AMS MSC:

20F55 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Reflection and Coxeter groups)

Pending Errata and Addenda

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