Let $G$ be a finite group and $d$ a divisor of $|G|$. Is there a subset $A$ of $G$ with $|A|=d$ such that $d$ divides order of the subgroup generated by $A$ ($|A|$ divides $|\langle A\rangle|$)?
Note that it is true if $G$ is generated by two elements, and also holds for all abelian (and CLT) groups.