 Finite groups of planar rotations are cyclic

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.