Is there any finite group $G$ of order $n=m^2$ satisfying the following conditions?
(a) There is no subgroup of order $m$;
(b) There exist subsets $A$, $B$ such that $|A|=|B|=m$ and $G=AB$.
$n$ must be $\geq 24^2=576$ ($m=24, 28, 30, \cdots$). Also, if answer of this problem is positive, then we obtain a counterexample for the problem Factorization of a finite group by two subsets.
Thanks in advance for your feedback.