Sylow p-subgroup: Understanding a proof

Question

I don't understand the part of the followng proof that I underlined with red. I don't see why $PN/N \cong P/(P \cap N)$ implies $PN/N$ is a p-subgroup of $G/N$.

Answer 1

A quotient of a $p$-group is a $p$-group. This applies in the infinite case, where we define a $p$-group to be one all whose elements have prime power order $p^k$, for some $k$.