These are concluding remarks from Dummit and Foote Abstract Algebra, Ch 3, sec 3:
We have, in the course of the proof of the isomorphism theorems, encountered situations where a homomorphism $\varphi$ on the quotient group $G/N$ is specified by giving the value of $\varphi$ on the coset $gN$ in terms of the representative $g$ alone. In each instance we then had to prove $\varphi$ was well defined, i.e., was independent of the choice of $g$. In effect we are defining a homomorphism, $\Phi$, on $G$ itself by specifying the value of $\varphi$ at $g$. Then independence of $g$ is equivalent to requiring that $\Phi$ be trivial on $N$, so that
- $\varphi$ is well defined on $G/N$ if and only if $N\le \ker \Phi$.
Although I don't quite understand what is going on in this part of the section, I would like to see a proof of bullet assertion. Here is my attempt:
$\implies$: Suppose $\varphi$ well defined then, and let $n\in N$, then $\Phi(n)=\varphi\circ\pi(n)=\varphi(nN)=\varphi(N)=e_H$.
But, I can't show reverse implication. I think that since we have not shown yet that $\varphi$ is indeed well defined, using $\Phi=\varphi\circ\pi$ will be illegal.
Also, if you can attach other reading or lecture note link, that discusses aforementioned topic in more detail, I will be obliged.