Let $G$ be a finite group such that $HK=KH$ for any subgroups $H,K$ of $G$ . Then is every subgroup of $G$ normal ?
[Edit] (to anyone thinking this question is not worth keeping: Here is the response from the OP to a comment from Tobias Kildetoft stating that the condition implies uniqueness of all Sylow subgroups, and hence that $G$ is nilpotent, JL)
If for a fixed prime $p$, $H,K$ are two different Sylow-p subgroups then $|H \cap K| < |H|=|K|$ and then the subgroup $HK$ is a $p$-subgroup with order $|H||K|/|H \cap K| > |H|$ , impossible ! Hence for a given prime $p$ , there is a unique Sylow-p subgroup . But I have no idea whether these line of arguments passes to all subgroups or not ...