If $G$ is a finite group such that $H,K\leq G$ and $HK\leq G$, can we say that $G$ is solvable?

Question

Question: Suppose that $H$ and $K$ are subgroups of a finite group $G$ such that $HK\leq G$.

I believe I have seen before that we can claim that $G$ is a solvable group, but I can't think about how to show it. Maybe since $HK\leq G$, then we can say, WLOG that $H\leq G$ and $K\trianglelefteq G$, and then try and play with the derived series?

Any help is greatly appreciated!

Answer 1

I interpret your question as follows: If $HK\le G$ for all $H,K\le G$, then $G$ is solvable? (assuming $|G|<\infty$) The answer is yes. Let $H$ and $K$ be Sylow $p$-subgroups of $G$. Then $HK\le G$ implies $H=K$. Hence, every Syow subgroup is normal and $G$ is nilpotent. In particular, $G$ is solvable.

(In general, groups with this property are called Iwasawa groups or sometimes modular groups (see Wikipedia)).