= Involutions in an even dihedral group
{title2=$r^n,\ sr^j\ (0\le j<2n)$}
For the <dihedral group> of a regular $2n$-gon, $n\ge2$, use $r^{2n}=s^2=1$ and $srs=r^{-1}$. Its <involutions> are the central half-turn $r^n$ and all reflections $sr^j$. The half-turn has singleton <conjugacy class> and <normal closure> $\{1,r^n\}$. Conjugating a reflection changes its exponent by an even integer or negates it, so its <conjugacy class> is $\{sr^{j+2k}:0\le k<n\}$. Products of consecutive reflections generate $r^2$, giving <normal closure> $\langle r^2,sr^j\rangle$, of order $2n$ and index two. The two reflection classes are distinguished by exponent parity.
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