Why is there a one-to-one correspondence between homomorphic images of a group $G$ and normal subgroups of $G$?

Question

I was reading about group theory in Herstein's book (mentioned below) and I came across a couple of propositions that were not clear to me, in the sense that I couldn't quite figure out why they were true and I would really appreciate if you could help me understand them.

They are the following:

Thus there is a one-to-one correspondence between homomorphic images of G and normal subgroups of G. [...] The set of groups so constructed yields all homomorphic images of G (up to isomorphisms).

Herstein, I. N., Topics in algebra, Lexington, TX: Xerox College Publishing. xi, 388 p. (1975). ZBL1230.00004.

The constructed set mentioned in the quote above is the set $ \left\{G/N: N \triangleleft G \right\} $

I can understand why given a normal subgroup $N$ of $G$, one can associate a homomorphic image of G, namely, $G/N$, but is the converse true, I mean, can I associate a normal subgroup $N$ of $G$ given a homomorphic image of a group $G$? How exactly does the given set yield all homomorphic images of $G$?

Thanks in advance!

Answer 1

The trick between the correspondence lies in the isomorphism theorems. You already know that to every normal subgroup of $G$ there corresponds a unique homomorphic image (upto isomorphism). Now, suppose that $h:G\to H$ is a group homomorphism, then $\ker(h)$ is normal, and by the first isomorphism theorem $$G/\ker(h)\cong h(G).$$

Answer 2

This is basically the First Ismorphism Theorem. Let's say $H$ is a homomorphic image of $G$, i.e. there exists a morphism $\varphi$ such that $\varphi : G \to H$ is surjective. Then $N:= \text{ker} \ \varphi$ is a normal subgroup of $G$ and $G/N \cong H$. Then the correspondence \begin{align} \{ \text{normal subgroups of } G \} &\to \{ \text{homomorphic image of } G \} \\ N &\mapsto G/N \end{align} is inverse to (up to isomorphism) \begin{align} \{ \text{homomorphic image of } G \} &\to \{ \text{normal subgroups of } G \} \\ H &\mapsto N:= \text{ker} \ \varphi, \text{ where } \varphi : G \to H. \end{align}