  Under suitable assumptions on the base field, we prove that a commutative
semisimple Yetter-Drinfel'd Hopf algebra over a finite abelian group is
trivial, i.e., is an ordinary Hopf algebra, if its dimension is relatively
prime to the order of the finite abelian group. Furthermore, we prove that a
finite-dimensional cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra
contains a trivial Yetter-Drinfel'd Hopf subalgebra of dimension greater than
one, at least if the Yetter-Drinfel'd Hopf algebra itself has dimension greater
than one.
