$r$-cycle to a power $k$ is also an $r$-cycle if and only if $\gcd(k, r) = 1$

Question

Let $\sigma$ be an $r$-cycle in $S_n$ and let $k\in\Bbb Z$. Show that $\sigma^k$ is also an $r$-cycle if and only if $\gcd(k,r)=1$.

Answer 1

Hint: $\left\langle \sigma^{k}\right\rangle =\left\langle\sigma^{gcd\left(k,r\right)}\right\rangle$, $ord\left(\sigma\right)=r$.