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 $$G / N$$.