Outer automorphism groups of finite p-nilpotent groups

When does a finite group possess a non-trivial outer automorphism? In this paper for a finite p-nilpotent group G we shall prove the following results. (a) If \O-p(G)\ > p, then p divides the order of Out(G). (b) If O-p(G) is non-abelian, then p divides the order of C-Out(G) (Z(G)). Our results extend a famous theorem of Gaschutz and can be applied to the investigation of finite complete groups.