On the conjecture of non-inner automorphisms of finite $p$-groups with a non-trivial abelian direct factor

TL;DR

The work addresses the Berkovich conjecture that every finite non-abelian $p$-group admits a non-inner automorphism of order $p$. It proves the conjecture for groups possessing a non-trivial abelian direct factor by constructing a central automorphism of order $p$ that fixes the Frattini subgroup $Φ(G)$. The automorphism is shown to be central and to fix $G'$, $G^p$, and hence $Φ(G)$, with extensions to products having a single abelian factor, yielding a broad Corollary for groups that are not purely non-abelian. An explicit example on a group of order $3^7$ confirms the existence of a non-inner central automorphism of order $3$ that fixes $Φ(G)$, illustrating the construction in a concrete setting.

Abstract

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that the conjecture is true when a finite non-abelian $p$-group $G$ has a non-trivial abelian direct factor. Moreover, we prove that the non-inner automorphism is central and fixes $Φ(G)$ elementwise. As a consequence, we prove that every group which is not purely non-abelian has a non-inner central automorphism of order $p$ which fixes $Φ(G)$ elementwise.

Theorems & Definitions (6)

• Lemma 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• proof
• Corollary 2.4