Non-inner automorphisms of order $p$ in finite $p$-groups admitting cyclic center

TL;DR

This paper proves that certain finite p-groups with cyclic center admit non-inner automorphisms of order p, under specific centralizer conditions, expanding understanding of automorphism structures in p-groups.

Contribution

It establishes a new criterion involving centralizers for the existence of non-inner automorphisms of order p in finite p-groups with cyclic center.

Findings

If the centralizer condition is met, then G has a non-inner automorphism of order p.

The result applies specifically to non-abelian p-groups with cyclic center and odd prime p.

Provides a new perspective on automorphism structure in p-groups.

Abstract

Let $G$ be a finite non-abelian $p$-group admitting cyclic center and $p$ be an odd prime. In this paper, we prove that if $C_{G}(Z(\gamma_{3}(G)G^{p}))\nleqslant\gamma_{3}(G)G^{p}$, then $G$ has a non-inner automorphism of order $p$.