= Conjugacy classes of a finite dihedral group
Use $D_n$ for the group of order $2n$ with rotation $R$ and reflection $m$. Every element is $R^j$ or $R^jm$, $0\leq j<n$. The rotational <conjugacy classes> are the sets $\{R^j,R^{-j}\}$, reduced to singletons for central rotations. For odd $n$, all reflections form one class, giving $(n+3)/2$ classes in total. For even $n$, the identity and $R^{n/2}$ are central singletons, the other rotations form pairs, and the reflections split into even and odd powers of $R$, giving $n/2+3$ classes. Indeed conjugation by $R^s$ changes a reflection exponent by $2s$, while conjugation by $m$ negates it. For odd prime $n$, a normal subgroup containing a reflection must contain all reflections and hence the whole group; the proper normal subgroups are therefore only the identity subgroup and the rotation subgroup.
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