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:

$P N / N \cong P / \paren {P \cap N}$

We have that:

$P N / N = \set {p N : p \in P}$

and so every element of $P N / N$ has order a power of $p$.

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

From Intersection of Normal Subgroup with Sylow P-Subgroup, we have that:

$p \nmid \index G {P N}$

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

$\blacksquare$