isomorphism between two factor groups

Question

We know that $A$ and $B$ are subgroups of $G$. Moreover, $B$ is a normal group of $G$. I've proved that if $BA=AB$ then $AB$ is a subgroup of $G$.

Then I have to prove that factor group $A/(A \cap B)$ and $BA/B$ are isomorphic.

I have shown that there are two natural homomorphisms $\pi: A \rightarrow BA/B$ and $\phi: A \rightarrow A/(A \cap B)$.

Could you give me a suggestion how to finish the proof?

Answer 1

Hint: It suffices to show that $\pi$ is onto and that $\ker(\pi)=A\cap B$ then the conclusion follows from the first isomorphism theorem.

Answer 2

If $B$ is normal, it is guaranteed that $AB$ and $BA$ are subgroups. You don't need to suppose that $AB=BA$. If you are using a textbook, you may wish to look in the section that deals with the Correspondence and Isomorphism Theorems. These results are usually established at that point.