Q: Product of three supersoluble subgroups I need a proof of the following statement:


*

*Let a finite group $G=AB=AC=BC$ be the product of three supersoluble groups $A$, $B$ and $C$. If the commutator subgroup $G'$ is nilpotent, then $G$ is supersoluble.


As a consequence of a Baer's result, in a finite group $G$ with nilpotent commutator subgroup, one can consider the largest normal and supersoluble subgroup $H$ of $G$; so, I think that I can prove the statement 1. by proving this other one:


*

*Let a finite group $G=AB$ be the product of two supersoluble subgroups and let the commutator subgroup $G'$ be nilpotent. If $H$ is the largest normal and supersoluble subgroup of $G$, then $A\leq H$ or $B\leq H$.


I thought that this could be the way, but I'm not sure. Can someone help me? Thank you very much.  
