18.112 (2014)

Is it true that the orders of all elements of a finite group $G$ are powers of primes if, for every divisor $d$ ($d > 1$) of $|G|$ and for every subgroup $H$ of $G$ of order coprime to $d$, the order $|H|$ divides the number of elements of $G$ of order $d$?

The converse is true (W. J. Shi, Math. Forum (Vladikavkaz), 6 (2012), 152–154).