Let $\phi : G_1 \rightarrow G_2$ be a surjective group homomorphism. Let $H_1$ be a normal subgroup of $G_1$ and suppose that $\phi (H_1) = H_2$. Prove or disprove that $G_1/H_1 \cong G_2/H_2$.
I say they are indeed isomorphic. Because:
Let $f$ be the group homomorphism from $G_1$ to $G_2/H_2$ that sends $a$ to $\phi(a)$. Then the kernel of $f$ is everything that is sent to $H_2$. Well by assumption this is $H_1$. Since $\phi$ is surjective, so is $f$, so by the first isomorphism theorem, $G_1/H_1$ is isomorphic to $G_2/H_2$
Is this correct reasoning?