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
We have that $P \le G$ and $ N \lhd G$.

So by the Second Isomorphism Theorem for Groups, we have:
 * $P N / N \cong P / \left({P \cap N}\right)$

Note that $P N / N$ is a $p$-subgroup of $G / N$, since $P N / N = \left\{pN : p\in P\right\}$, and every element has order a power of $p$.

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 $G / N$.