Finite $p$-groups with non-cyclic center have non-inner automorphism of order $p$

TL;DR

This paper investigates the automorphism structure of finite p-groups, showing that those with a non-cyclic center necessarily possess a non-inner automorphism of order p, revealing new insights into their symmetry properties.

Contribution

It establishes a new criterion linking the non-cyclicity of the center of a p-group to the existence of specific automorphisms of order p.

Findings

Finite p-groups with non-cyclic center have non-inner automorphisms of order p.

The result deepens understanding of automorphism groups in p-group theory.

Provides a characterization connecting the center's structure to automorphism properties.

Abstract

Let $G$ be a finite $p$-group.