A, B subgroups of G, B/A abelian. Show that BN/AN is abelian. I'm going through a past paper on Group theory and I wondered if someone could help with a solution to this.
Let $A$, $B$ be subgroups of $G$ such that $A\triangleleft B$ and $B/A$ is abelian. Also $N\triangleleft G$. I proved that $AN\triangleleft BN$ and now I need to show that $BN/AN$ is abelian.
Thanks

By the Isomorphism Theorems, since $A\subset B$ so $BA=B$, we have:
$$\frac{BN}{AN} = \frac{B(AN)}{AN} \cong \frac{B}{B\cap AN}.$$
Since $A\leq AN$, then $A = B\cap A\leq B\cap AN$. Hence
$$\frac{B}{B\cap AN}$$ is a quotient of $B/A$, hence abelian since $B/A$ is abelian.
