Relation between set of Normal subgroups and set of Homomorphism images of a group $G$

Question

let $G$ be any group.let $A$ is set of all normal subgroups of $G$, and $B$ is set of all homomorphic images of $G$.

Is there any type of relation between these sets.Mean some type of bijection like that.

For any normal subgroup $N \in A$ there is a homomorphic image $G/N \in B$ and conversely for any homomorphic image $G' \in B$ there is normal subgroup $N=\ker\phi \in A$ where $\phi$ is homomorphism from $G$ to $G'$

Is this a bijection between A and B ?? or i am doing something wrong . please help.

Answer 1

Yes.

Let $G$ be a group. Given a normal subgroup $N$ it is well known that $G/N$ is a group and $\varphi:G\rightarrow G/N$ taking $g$ to $gN$ is a homomorphism whose kernel is $N$.

On the other hand if $\varphi:G\rightarrow H$ is a surjective homomorphism (so $H$ is an homomorphic image of $G$. Then by the first isomorphism theorem $N:=\ker \varphi$ is a normal subgroup of $G$ such that $H=G/N$.