University of Isfahan International Journal of Group Theory 2251-7650 8 1 2019 01 13 On noninner automorphisms of finite $p$-groups that fix the center elementwise 1 9 22412 10.22108/ijgt.2018.108082.1457 EN S. Mohsen Ghoraishi Shahid Chamran University Journal Article 2017 11 19 In this paper we show that every finite nonabelian $p$-group $G$ in which the Frattini subgroup $\Phi(G)$ has order $\leq p^5$ 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 $p^8$. https://ijgt.ui.ac.ir/article_22412_bb6fd44e271320636320a62cf00c7ba2.pdf