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 $\Phi(G)$ has order $\leq p^۵$ 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 $p^۸$.

Keywords:

$p$-groups , automorphisms , noninner automorphisms