Depth two Hopf subalgebras of a semisimple Hopf algebra

A depth two Hopf subalgebra K of a semisimple Hopf algebra H is shown to be a normal Hopf subalgebra. On the one hand, we prove this using Galois theory of quantum groupoids. On the other hand, we give a second proof using character theory of semisimple Hopf algebras.