Let $A_1$ and $A_2$ be a group and $\phi: A_1 \to A_2$ be a surjective group homomorphism. Also, $G_1 \unlhd A_1$ and $\phi(G_1) = G_2$. I have to prove/disprove this statement: Is $A_1/G_1 \cong A_2/G_2$?
My claim is that the statement is true. I first showed that if $G_1 \unlhd A_1$, then $G_2 \unlhd A_2$. Then, I defined two maps: $f:A_1/G_1 \to A_2$, which is an isomorphism by first-isomorphism theorem. And then, another map $j:A_2 \to A_2/G_2$, which is a surjective group homomorphism. I thought that this would work, but without being able to prove that $j$ is injective, I cannot show that $j$ is a bijection. Can anyone check my work? Or is the statement false?