Proving that a finite rotational group in $\mathbb{R}^2$ is cyclic

Question

G is a finite set of rotations about the origin in $\mathbb{R}^2$, closed under composition. I believe I can prove that G forms an abelian group under composition, but how could I go about proving that G is cyclic?

Answer 1

The remainder upon division of $\theta '$ by $\theta$ is $\alpha$. So $\theta '=q\theta +\alpha$, where $\alpha\lt\theta $, by the division algorithm. But $\alpha\in G$ and $\alpha\lt\theta\implies \alpha=0$, by minimality of $\theta$.

Answer 2

It's not true that a finite group of rotations is cyclic. The finite groups of rotations that preserve an icosahedron is not even solvable. (It's isomorphic to $A_5$.)

Possibly you should focus on proving that a finite group of rotations in the plane is cyclic.