On semisimple Hopf algebras of dimension pq

We consider the problem of the classification of semisimple Hopf algebras A of dimension pq where p < q are two prime numbers. First we prove that the order of the group of grouplike elements of A is not q, and that if it is p, then q = 1 (mod p). We use it to prove that if A and its dual Hopf algebra A* are of Frobenius type, then A is either a group algebra or a dual of a group algebra. Finally, we give a complete classification in dimension 3p, and a partial classification in dimensions 5p and 7p.