Dihedral group

A regular \(n\)-gon has \(n\) rotations, \(e,r,\dots,r^{n-1}\), and \(n\) reflections, \(s,sr,\dots,sr^{n-1}\), so \(|D_n|=2n\). Every element is written uniquely as \(r^k\) or \(sr^k\) with \(0\le k<n\). The rotations form the cyclic subgroup \(\langle r\rangle\), of index \(2\) and hence normal. Some authors write \(D_{2n}\) for this same group, of order \(2n\); check each book's convention.

The computation rule is \(sr^k=r^{-k}s\): moving a reflection across a rotation inverts the rotation. That is why \(D_n\) is not abelian for \(n\ge3\), while \(D_2\), the Klein group, is. Every reflection has order \(2\), and \(r^k\) has order \(n/\gcd(n,k)\).

Guiding example: \(D_3\) is the symmetry group of an equilateral triangle; it permutes the three vertices and is identified with \(S_3\). \(D_4\), of order \(8\), has center \(\{e,r^2\}\); in general the center is trivial if \(n\) is odd and equals \(\{e,r^{n/2}\}\) if \(n\) is even. A typical mistake is forgetting that \(rs\ne sr\): the order of the factors matters.

Every dihedral group is a subgroup of \(S_n\), since each symmetry permutes the vertices. Together with the symmetric and alternating groups it serves as a testing ground for normal subgroups, quotients and actions, and it reappears later in the classification of groups of small order.