Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of $G$ such that $P = (P \cap H)(P \cap K)$.
It is not hard to prove that there exist $p$-Sylow subgroups of $H$ and $K$, $P_1$ and $P_2$ respectively, such that $P_1 \cap P_2$ is a $p$-Sylow subgroup of $H \cap K$, but I cannot guarantee that $P_1P_2$ is a subgroup of $G$ (and thus the $p$-Sylow of $G$ that we are looking for). Is there any way to complete the proof or it is not the right way?