I'd appreciate it if you consider this question and together with its hint:
Let $G=AB$ be a finite group which is the internal direct product of $A$ and $B$ which are non-abelian simple groups. Show that the only proper, nontrivial normal subgroups of $G$ are $A$ and $B$.
Hint: If $N$ is another such subgroup, we would have to consider the commutator $[N,A]$.
In fact I am not sure how to use this hint!