Let $f$ be a homomorphism mapping $G$ to $J$ groups. Assume that $J$ is abelian. Prove that if $H$ is a subgroup of $G$ and if $\mathrm{ker}(f)$ is a subset of $H$, then $H$ is normal in $G$.
I honestly have no idea how to start this proof. The fact that $J$ is abelian gives me that for all $a$ and $b$ in $G$, $f(ab) = f(ba)$; but this tells me nothing since $f$ is not injective. Furthermore, what exactly can I get from the fact that $\mathrm{ker}(f)$ is a subset of $H$? I understand the given information but I am having issues putting it all together as I see no possible relation!
My plan was to piece the information all together to somehow set things up for the normal subgroup test, however since I see no connection with the given information my approach seems to be rather futile.
Another thought I just got was to prove that $f$ is injective then use the fact that $J$ is abelian to show that $ah = ha$ for all $a$ in $G$ and $h$ in $H$.
Thanks.