I'm reading Robert Ash's Basic Abstract Algebra. Here:
I've tried to find the proof that every normal subgroup is the kernel of a homomorphism before looking in the book. I got stuck thinking about an $f:G\to H$ and failing, eventually I gave up.
When I looked at the proof, it was easy to follow but he picks $\pi: G \to G/N$ and I guess I'd never have guessed that. I am confused with the following: I was expecting the result to be valid for any homomorphism but he picks that exact one. My doubt may be a little bit silly, but why this proof for $\pi:G\to G/N$ makes the result valid for any $H\neq G/N$? I guess It's clear that that proposition depends only on $G$ and hence, we can pick any H but I am unsure why this is legitimate.