Noninner Automorphisms of Order p of Finite p-Groups

The main result of this paper shows that if G is a finite nonabelian p-group and if CG(Z((G))) ≠ (G), then G has a noninner automorphism of order p which fixes (G). This reduces the verification of the longstanding conjecture that every finite nonabelian p-group G has a noninner automorphism of order p to the degenerate case in which CG(Z((G))) = (G).