on derived group of two cyclic subgroups

Question

Let $G=AB$ be a finite group such that $A,B$ are cyclic. Can i conclude that $G^{\prime}$ is cyclic?

My try: Clearly, $G^{\prime}=[A,B]$. Let $A=\langle a \rangle$ and $B=\langle b \rangle$. I think $G^{\prime}=\langle[a,b]\rangle$. is it true?

Answer 1

Counterexample: $S_3\times S_3$

Let $A=\langle((1,2,3),(1,2))\rangle$ and $B=\langle ((1,2),(1,2,3)\rangle$. Then $G=AB$ with $A$, $B$ cyclic as required.

However $[G,G]=[S_3,S_3]\times[S_3,S_3]=C_3\times C_3$ is not cyclic.