On noninner automorphisms of finite p -groups that fix the center elementwise

In this paper we show that every finite nonabelian p-group G in which the Frattini subgroup Φ(G) has order ≤p^5^ admits a noninner automorphism of order p leaving the center Z(G) elementwise fixed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least p8.