= Finite groups of planar rotations are cyclic
{title2=$G=\langle R_{2\pi/m}\rangle$}
A finite <group> of <planar rotations> about one point is a <cyclic group>. For a nontrivial group, choose its smallest positive rotation angle $\alpha$. Subtracting multiples of $\alpha$ from any other group angle leaves a group angle in $[0,\alpha)$, hence zero. Applying the same argument to $2\pi$ shows that $\alpha=2\pi/m$ for a positive integer $m$. The identity-only group is cyclic as well.
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