Group Epimorphism Induces Bijection between Subgroups

Theorem
Let $G_1$ and $G_2$ be groups whose identities are $e_{G_1}$ and $e_{G_2}$ respectively.

Let $\phi: G_1 \to G_2$ be a group epimorphism.

Let $K := \map \ker \phi$ be the kernel of $\phi$.

Let $\mathbb H_1 = \set {H \subseteq G_1: H \le G_1, K \subseteq H}$ be the set of subgroups of $G_1$ which contain $K$.

Let $\mathbb H_2 = \set {H \subseteq G_2: H \le G_2}$ be the set of subgroups of $G_2$.

Then there exists a bijection $Q: \mathbb H_1 \leftrightarrow \mathbb H_2$ such that:
 * $\forall N \lhd G_1: \map Q N \lhd G_2$
 * $\forall N \lhd G_2: \map {Q^{-1} } N \lhd G_1$

where $N \lhd G_1$ denotes that $N$ is a normal subgroup of $G_1$.

That is, normal subgroups map bijectively to normal subgroups under $Q$.

Proof
Let $Q$ be the mapping defined as:
 * $\forall H \le \mathbb H_1: \map Q H = \set {\map \phi h: h \in H}$

Let $H$ be a subgroup of $G_1$ such that $K \subseteq H$.

From Group Homomorphism Preserves Subgroups, $\phi \sqbrk H$ is a subgroup of $G_2$.

This establishes that $Q$ is actually a mapping.

Let $N \lhd G_1$.

From Group Epimorphism Preserves Normal Subgroups, $\phi \sqbrk N$ is a normal subgroup of $G_2$.

This establishes that:
 * $\forall N \lhd G_1: \map Q N \lhd G_2$

Next it is shown that $Q$ is a bijection.

Injective Nature of $Q$
Let $H, J \in \mathbb H_1$.

Let $\map Q H = \map Q J$.

Let $h \in H$.

A similar argument shows that $J \subseteq H$.

So by definition of set equality:
 * $H = J$

Thus:
 * $\map Q H = \map Q J \implies H = J$

So by definition, $Q$ is injective.

Surjective Nature of $Q$
Now let $N' \in \mathbb H_2$.

By definition of $\mathbb H_2$, $N'$ is a subgroup of $G_2$.

Let $N = \set {x: \map \phi x = N'}$.

We have from Identity of Subgroup that $e_{G_2} \in N'$.

Thus by definition of kernel, $K \subseteq N$.

Now suppose $\map \phi x, \map \phi y \in N'$.

Then:

So by the One-Step Subgroup Test, $N$ is a subgroup of $G_1$.

It has been established that $K \subseteq N$, and so $N \in \mathbb H_1$.

Thus it follows that for all $N' \in \mathbb H_2$ there exists $N \in H_1$ such that $\map Q N = N'$.

So $Q$ is a surjection.

So $Q$ has been shown to be both an injection and a surjection, and so by definition is a bijection.

Finally, it can then be shown that if $N'$ is normal in $G_2$, it follows that $N = \map {Q^{-1} } {N'}$ is normal in $G_1$.

This establishes that:
 * $\forall N \lhd G_2: \map {Q^{-1} } N \lhd G_1$

Also see

 * First Isomorphism Theorem