Let $\displaystyle \rho \in Sym(n)$ and let p be a prime number. Show that $\displaystyle \rho^p = \iota$ if and only if the cycles of $\displaystyle \rho$ have lengths 1 or p.

I realise that this is an if and only if question, so it will need to be shown in the two directions.

For showing that if the cycles all have lengths 1 or p, then $\displaystyle \rho^p = \iota$ I have have shown it for length 1 but can't see how to show it for p. I can't think of where to start for the proof in the other direction. Help anyone?