Is $HN/N$ equal to $H/N$?

Question

The question might be stupid but I was wondering if $(HN)/N = H/N$. It seems true, but in the second isomorphism theorem, one can show that $H/(H\cap N)$ is isomorphic to $HN/N$. So it's weird if the former is true.

Answer 1

Well, yes. Let's suppose $H\leq G, N\trianglelefteq G$. We can define a projection homomorphism $\pi:G\to G/N$ by $g\to gN$. Then it is true that $\pi(H)=\pi(HN)$. So what you can "imagine" as $H/N$ is really equal to $HN/N$ as a set of elements. But in order to think of it as a quotient group we want $N$ to be contained in $H$. Since it is not always the case we talk about $HN/N$ instead.