Remark

The converse does not always hold. That is, there are finite groups G with divisors d of |G| for which there is no element of order d.

For example, there is no element of order 15 in S₅, while clearly 15 divides 120, the order of S₅. To see that there is no such element, note that the disjoint cycles notation of an element of order 15 should either have a 15-cycle or cycles of orders 3 and 5; both alternatives are impossible as there are only 5 letters.