  A finite group is said to have "perfect order classes" if the number of
elements of any given order is either zero or a divisor of the order of the
group.
  The purpose of this note is to describe explicitly the finite Hamiltonian
groups with perfect order classes. We show that a finite Hamiltonian group has
perfect order classes if, and only if, it is isomorphic to the direct product
of the quaternion group of order $8$, a non-trivial cyclic $3$-group and a
group of order at most $2$.
  Theorem. A finite Hamiltonian group has perfect order classes if, and only
if, it is isomorphic either to $Q\times C_{3^k}$ or to $Q\times C_{2}\times
C_{3^k}$, for some positive integer $k$.
