dihedral group (original) (raw)

The nth 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 by𝒟n. An alternate notation is 𝒟2⁢n; in this approach, the subscript indicates the order of the group.

Lettingω=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 identity) are given by

and the reflections Mk,k=0,…,n-1 by

The abstract group structure is given by

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 𝒟n is a rank-1 Coxeter group.

Since the group acts by linear transformations

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

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

the latter polynomial being the real part 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.