# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1702.02293