If $N$ and $M$ are normal subgroups and $N$ and $M$ have no common element other than $e$ then prove that for all $m \in M$ and $n\in N$, $mn=nm$. My approach ; I proved the $MN$ to be a normal sub group whence $mn=nm$.

Try showing $n^{-1}m^{-1}nm\in N\cap M$.