Analogous of Fitting theorem for abelian group is true?

Question

If $M$ and $N$ are $2$ normal abelian subgroups of a group $G$, then $MN$ is abelian?

I think that response is negative but i don't found a easy counterexample.

Answer 1

Let $G=D_4$ with presentation $\langle a, b \mid a^4=b^2=e, ab=ba^{-1} \rangle$. Let $M= \langle a \rangle, N = \langle a^2, b \rangle$.