Let $G$ be a group and $P$ a $p$-Sylow. I have to show that there exists a $g \in G$ such that for $H \leq G$, $ gPg^{-1} \cap H$ is a $p$-Sylow of $H.$
If $H$ is normal is easy, but when I consider the case in which $H$ is not normal I'm stuck. I thought that maybe all the elements in $H$ of order $p^{\alpha}$ must be in som $p$-Sylow but I haven't been able to prove that. Also I tried to prove it using the product fromula but it didn't work at all.
Please help.