Order of a permutation divides n in Sn

Question

Let $\theta \in S_n$, and for any $k \in \mathbb{N}$, either $\theta^k = I_{I(n)}$ or $\theta^k$ has no fixed elements. Show that $o(\theta) | n$.

$I_{I(n)}$ denotes the identity.

I'm completely stuck on this problem.

Answer 1

Hint: write $\theta$ as a product of cycles with disjoint support. Let $c_1\dots c_p$ be the length of the cycles. Translate the hypothesis about $\theta$ in terms of $c_1\dots c_p$.