On noninner automorphisms of finite $p$-groups that fix the center elementwise

Document Type: Research Paper

Author

Shahid Chamran University

Abstract

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 $p^8$.

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