I am thinking on this problem:
If $N\lhd H×K$ then either $N$ is abelian or $N$ intersects one of $H$ or $K$ nontrivially.
I assume; $N$ is not abelian so, there is $(n,n')$ and $(m,m')$ in $N$ such that $([n,m],[n',m'])\neq 1$. But I can’t go further. Hints are appreciated. Thanks.