On automorphisms of finite p-groups with small Frattini subgroup

A longstanding conjecture asserts that every finite nonabelian p-group G has at least one noninner automorphism of order p. 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. As a consequence it follows that the order of a possible counterexample to the conjecture is at least p^8.