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

Abstract

In this paper, we prove that if [Formula: see text] is a finite nonabelian [Formula: see text]-group with [Formula: see text] and [Formula: see text], then [Formula: see text] has a noninner automorphism of order [Formula: see text], where [Formula: see text] is the third member of the upper central series of [Formula: see text] and [Formula: see text] is the minimal number of generators of [Formula: see text]. This reduces the verification of the longstanding conjecture that every finite nonabelian [Formula: see text]-group [Formula: see text] has a noninner automorphism of order [Formula: see text] to the case in which [Formula: see text] for [Formula: see text]. Moreover, as a consequence, we prove that every finite [Formula: see text]-group of order less than [Formula: see text] has a noninner automorphism of order [Formula: see text].