Some properties of finite-dimensional semisimple Hopf algebras

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dim V divides dim H) [Ka]. It was proved that the conjecture is true for H of dimension p n , p prime [MW], and that if H has a 2-dimensional representation then dim H is even [NR]. It was also proved that the dimension of an irreducible