On the conjecture of non-inner automorphisms of finite $p$-groups

TL;DR

This paper proves the existence of non-inner automorphisms of order p in certain finite non-abelian p-groups, advancing the understanding of automorphism structures in these groups.

Contribution

It establishes conditions under which a finite non-abelian p-group has a non-inner automorphism of order p fixing the Frattini subgroup.

Findings

Proves existence of non-inner automorphisms in specific p-groups.

Identifies structural conditions involving maximal subgroups and centers.

Advances the conjecture for a class of monolithic p-groups.

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 if $G$ is an odd order finite non-abelian monolithic $p$-group such that every maximal subgroup of $G$ is non-abelian and $[Z(M), g] \leq Z(G)$ for every maximal subgroup $M$ of $G$ and $g \in G \setminus M$. Then $G$ has a non-inner automorphism of order $p$ leaving the Frattini subgroup $\Phi(G)$ elementwise fixed.