Properties of a permutation matrix
Hint: By the pigeonhole principle, there exist integers $m<n$ such that $P^m=P^n$. Conclude that we necessarily have $P^{n-m}=I$.
Hint: By the pigeonhole principle, there exist integers $m<n$ such that $P^m=P^n$. Conclude that we necessarily have $P^{n-m}=I$.