  Recently, important progress has been made in the study of finite-dimensional
semisimple Hopf algebras over a field of characteristic zero. Yet, very little
is known over a field of positive characteristic. In this paper we prove some
results on finite-dimensional semisimple and cosemisimple Hopf algebras A over
a field of positive characteristic, notably Kaplansky's 5th conjecture on the
order of the antipode of A. These results have already been proved over a field
of characteristic zero, so in a sense we demonstrate that it is sufficient to
consider semisimple Hopf algebras over such a field (they are also
cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for
semisimple and cosemisimple Hopf algebras over a field of positive
characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments
of deformation theory from positive to zero characteristic. The key ingredient
of the proof is the theorem that the bialgebra cohomology groups of A vanish.
