Prove the group of rotations of $\mathbb R^2$ about origin is cyclic I am self-studying group theory through the text 'Algebra and Geometry' by A.F.Beardon. I am struggling at a problem which states:

Show that a finite group of rotations of $\mathbb R^2$ about the origin is a cyclic group. Construct a proper subgroup of the group of rotations of $\mathbb R^2$ about the origin that is not cyclic

So my questions are :


*

*What is meant by a finite group of rotations of $\mathbb R^2$ about origin?

*How to decide which rotation group is non-cyclic?

*How to solve this problem?
 A: As @Qiaochu Yuan said :
1) It means a finite subgroup of the group $$R:=\Big\{r_\theta=\begin{pmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)\end{pmatrix};\theta\in\mathbb{R}\Big\}$$ which is itself a subgroup of $GL_2(\mathbb{R}),$
2) For example, $R$ is uncountable and so it is non-cyclic (each cyclic group is countable because you have the surjective morphism $\mathbb{Z}\to<g>$ defined by $n\mapsto g^n$),
3) Let $G=\{Id=r_0,r_{\theta_1},...,r_{\theta_n}\}$ and $\theta:=\min(\theta_1,...,\theta_n)\in]-\pi,\pi[,$ and show that $G=<r_\theta>.$ Let $r_{\theta'}\in G\backslash\{r_0\}.$ We can suppose that $\theta'\in]0,\pi[$ (replacing $r_{\theta'}$ with $r_{\theta'}^{-1}=r_{-\theta'}$). By definition, $\theta\leq \theta'.$ Let $q$ be the smaller integer such as $\theta'\geq q\theta$ and $\theta'<(q+1)\theta.$ Let $\alpha:=\theta'-q\theta,$ it is the angle of $r_{\theta'}\circ r_\theta^{-q}\in G$ and by minimality of $\theta$ you get $\alpha=0.$ So $r_{\theta'}=r_\theta^q$ and you get $G=<r_\theta>.$
A: For the non-cyclic case take the group $G$ generated by a rotation of $1$ radian. Then all elements of this group are of the form $e^{in}$ with $n\in\mathbb Z$and since $\pi$ is irrational there is no $n$ such that $e^{in}=e^{i2\pi k}$. Therefore, $G$ is acyclic.
A: If $G$ is a finite subgroup of rotations then
there must be an element 
$
\left(\begin{array}{cc}
\cos A&-\sin A\\
\sin A&\cos A
\end{array}\right)
\in G$ 
with $A$ as the minimal angle on the range $[0,\pi]$ and among the finite elements of $G$. 
One can use this element to generate all of $G$.
