Corollary 3

(a) Let $H$ be a subgroup of $G$. For every Sylow $p$-subgroup $P$ of $H$ there exists a Sylow $p$-subgroup $Q$ of $G$ such that $P = Q \cap H$.

(b) Conversely, if $Q$ is a Sylow $p$-subgroup of $G$ and $H$ is normal in $G$, the group $Q \cap H$ is a Sylow $p$-subgroup of $H$.

(a) The $p$-group $P$ is contained in a Sylow $p$-subgroup $Q$ of $G$ and $Q \cap H$ is a $p$-subgroup of $H$ containing $P$ and is hence equal to $P$.

(b) Let $P'$ be a Sylow $p$-subgroup of $H$. There exists an element $g \in G$ such that $gP'g^{-1} \subset Q$. As $H$ is normal, $P = gP'g^{-1}$ is contained in $H$ and hence in $Q \cap H$. As $Q \cap H$ is a $p$-subgroup of $H$ and $P$ is a Sylow $p$-subgroup of $H$, $P = Q \cap H$.