I realize this question has been asked 3 times now on the site, but I cannot understand the first line of the most popular proof:
Homomorphic image of a Sylow p-subgroup is Sylow p-subgroup.
The claim is as follows: Let $\phi: G\to H$ be a homomorphism of groups and let $P\leq G$ be a Sylow $p$-subgroup of $G$. Then $\phi(P)\leq H$ is a Sylow $p$-subgroup of $H$.
The first line of the proof states: $\phi(P)=\frac{PN}{N}$.
How do I understand this part? I would think to define $PN=\{ pn\in G : p\in N , n\in N\}$ but this makes no sense because $\phi(P)\subset H$ as sets.
Any ideas on how to understand the first line appreciated! (so I can move on to the second line).
This is a bit of "abuse of notation".
First of all, $N = \ker \phi$ (you should have stated this, but I got it from the linked post). This means that $H \cong G/N$ in a natural way (the first isomorphism theorem), so $\phi(P)$, while it is formally a subset of $H$, can be treated as a subset of $G/N$ instead via this isomorphism. Since $PN/N$ is also a subset of $G/N$, the equality then makes sense.