this question was posted on aops about 3 weeks ago, but never received an answer. Maybe it will fare better here?
Let $G$ be a finite group and $H$ a subgroup. Let $P_H$ be a $p$-Sylow subgroup of $H$. Prove that there exists a $p$-Sylow subgroup $P$ of $G$ such that $P_H=P\cap H$.
Source: The exercise comes from the first chapter of Lang. Thanks.