Given a permutation matrix $P$, is it true that there exist a natural number $n$ such that if $P$ is multiplied $n$ times, it gives the identity matrix (i.e., $\underbrace{P \times P \times \ldots \times P}_{n ~\text{times}} = I$)?
If so, I am looking for a proof and possibly an efficient way to find out the smallest soltion for $n$. Otherwise, a counterexample will be appreciated.