# prove that every Sylow p-subgroup of $S_n$ is abelian if and only if $n<p^2$

I was given the following problem:

Let $p$ be a prime, and $P$ be a Sylow p-subgroup of $S_n$. Prove that $P$ is abelian if and only if $n<p^2$

and I was also given the following hint:

find a subgroup of $S_{n^2}$ of order $p^p$, and show that no other element of $S_{p^2}$ commutes with all of it's elements

I do not understand how to tackle this problem, neither how the existence of such subgroup as mentioned in the hint helps.
