Isomorphism between quotient groups with normal subgroups

Question

I'm looking at a problem in my textbook and it says:

Let $ψ : G_1 → G_2$ be a surjective group homomorphism. Let $H_1$ be a normal subgroup of $G_1$ and suppose that $ψ(H_1) = H_2$. Prove or disprove that $G_1/H_1$ is isomorphic $G_2/H_2$.

So far, I know that since $H_1$ is normal in $G_1$, $H_2$ is also normal in $G_2$. I'm unsure how to proceed from here though, can someone give me a hint?

Answer 1

The statement is wrong. Just take $G_1=\mathbb{Z}_2$, $H_1=\{0\}$, $G_2=\{0\}$. Then $H_2=\{0\}$ and $G_1/H_1=\mathbb{Z}_2$ is not isomorphic to $G_2/H_2=\{0\}$.

Answer 2

This only holds if $ψ^{-1}(H_2)=H_1$, then use the isomorphism theorem:

$G_1 \to G_2 \to G_2/H_2$ is surjective. The kernel of the composition is then $H_1$. Now apply the isomorphism theorem and you're done.

Otherwise you could take for example $Z\times Z \to Z$ by $(1,1)\mapsto 1$ then the kernel is generated by $(1,-1)$ and you could take $H_1 = 0 \times Z$, but $Z \neq 0$.