= Finite subgroups of the plane orthogonal group
A finite <subgroup> of $O(2)$ is cyclic or dihedral. Its orientation-preserving part consists of <rotations> and is cyclic: choose the smallest positive rotation angle, divide any other angle by it with remainder, and use minimality to force zero remainder. The smallest angle is therefore $2\pi/n$ for some $n$. A trivial rotational subgroup is included by $n=1$.
If there is no orientation-reversing element, this is the whole <group>. Otherwise choose a <reflection> $s$. Every orientation-reversing element is $r^js$, where $r$ generates the rotations; these form the second <coset>. Since $srs=r^{-1}$, the group is a <dihedral group> of order $2n$. Its involutions are all $n$ reflections and, when $n$ is even, the extra half-turn $r^{n/2}$. Consequently it has $n$ order-two <cyclic subgroups> for odd $n$, and $n+1$ for even $n$.
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