Sylow subgroup of a subgroup from double cosets
= Sylow subgroup of a subgroup from double cosets
{c}
If $P$ is a <Sylow subgroup> of a finite group $G$ and $H\leq G$, some intersection $H\cap xPx^{-1}$ is a Sylow subgroup of $H$. Otherwise every double-coset size
$$
|HxP|=\frac{|H||P|}{|H\cap xPx^{-1}|}
$$
would be divisible by a larger power of $p$ than $|G|$, as would their sum $|G|$.