dihedral group

The nth dihedral groupMathworldPlanetmath is the symmetry group of the regularPlanetmathPlanetmath n-sided polygon. The group consists of n reflectionsMathworldPlanetmathPlanetmath, n-1 rotations, and the identity transformation. In this entry we will denote the group in question by π’Ÿn. An alternate notation is π’Ÿ2⁒n; in this approach, the subscript indicates the order of the group.

Letting Ο‰=exp⁑(2⁒π⁒i/n) denote a primitive nth root of unity, and assuming the polygon is centered at the origin, the rotations Rk,k=0,…,n-1 (Note: R0 denotes the identityPlanetmathPlanetmathPlanetmathPlanetmath) are given by


and the reflections Mk,k=0,…,n-1 by


The abstract group structure is given by

Rk⁒Rl =Rk+l, Rk⁒Ml =Mk+l
Mk⁒Ml =Rk-l, Mk⁒Rl =Mk-l,

where the addition and subtraction is carried out modulo n.

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


This means that π’Ÿn is a rank-1 Coxeter groupMathworldPlanetmath.

Since the group acts by linear transformations


there is a corresponding action on polynomialsMathworldPlanetmathPlanetmathPlanetmath p→p^, defined by


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


the latter polynomial being the real partMathworldPlanetmath of (x+i⁒y)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⁒π/n radians, and is also invariant with respect to complex conjugation. These two transformations generate the nth 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