Quotient of Sylow P-Subgroup

Theorem
Let $P$ be a Sylow $p$-subgroup of a finite group $G$.

Let $N$ be a normal subgroup of $G$.

Then $P N / N$ is a Sylow $p$-subgroup of $G / N$.

Proof
As $P \leq G$ and $ N \triangleleft G$, we have $P N / N \cong P / \left({P \cap N}\right)$ by the Second Isomorphism Theorem. Note that $P N$ is a $p$-subgroup of $G / N$.

From Intersection of Normal Subgroup with Sylow P-Subgroup,we have that $p \nmid \left[{G : P N}\right]$.

So $P N / N$ is a Sylow $p$-subgroup of $G / N$.