Let $G$ be a finite group such that $p\ |\ o(G)$ and for any two elements $a,b\in G$, $(ab)^p =a^p b^p$. Let $P$ be Sylow $p$-subgroup of $G$. Prove that there is a normal subgroup $N$ of $G$ such that $P \cap N= \{e\}$ and $PN=G$.
I think this assumption would lead us to the fact that Sylow $p$-subgroup of $G$ is unique. Once this is done, the rest trivially follows if we consider $N=G/P$. I'm stuck while doing that part. Am I in the correct direction?