A note on the existence of noninner automorphisms of order p in some finite p-groups

In this paper, we prove that if G is a finite nonabelian p-group with p > 2 and vertical bar Z(3)(G)/Z(G)vertical bar <= p(d(G)+1), then G has a noninner automorphism of order p, where Z(3)(G) is the third member of the upper central series of G and d(G) is the minimal number of generators of 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 case in which vertical bar Z(3)(G)/Z(G)vertical bar > p(d(G)+1) for p > 2. Moreover, as a consequence, we prove that every finite p-group of order less than p(9) has a noninner automorphism of order p.