Having some doubts proving exercise statement from Pinter's book. Here's quote:
Let $\alpha $ be a cycle of length $s$, say $\alpha = ( a_1, a_2 ... a_s )$. Prove that, if $s$ is a prime number, every power of $\alpha$ is a cycle.
I know well that power $k$ of cycle of length $k \cdot t$ is a product of $k$ disjoint cycles of length $t$. Prime number length will not let this happen with any power. But I'm not sure whether powering the cycle to a divisor of its length is the only way to break cycle apart.
Thank you for your attention.