  We prove a variety results on tensor product factorizations of finite
dimensional Hopf algebras (more generally Hopf algebras satisfying chain
conditions in suitable braided categories). The results are analogs of
well-known results on direct product factorizations of finite groups (or groups
with chain conditions) such as Fitting's Lemma and the uniqueness of the
Krull-Remak-Schmidt factorization. We analyze the notion of normal (and
conormal) Hopf algebra endomorphisms, and the structure of endomorphisms and
automorphisms of tensor products. The results are then applied to compute the
automorphism group of the Drinfeld double of a finite group in the case where
the group contains an abelian factor. (If it doesn't, the group can be
calculated by results of the first author.)
