Let $G$ be a finite group, and $p$ a prime. Let $P \in \operatorname{Syl}_p(G)$, and let $N$ be a normal subgroup of $G$. Use the conjugacy part of Sylow's theorem to prove that $P \cap N$ is a Sylow $p$-subgroup of $N$. Deduce that $PN/N$ is a Sylow $p$-subgroup of $G/N$.
How do we know that N will contain p-subgroup of G of same order as P ? I don't understand this question could somebody explain more.