Let $C$ be the symmetric group for the circle in $\mathbb{R}^2$. Show that for every $n \in \mathbb{N}$ there exists a $c \in C$ for which $|c|= n$.
The symmetric group of the circle is $\{e^{\frac{2k\pi i}{n}}\}$ where $i^2=-1,k\in\mathbb{N}_{\geq0},n\in\mathbb{N}_{\geq1}$? Also the symmetric group is infinite so there should be such $c$. I would need some hints on how to approach the proof?