On Central Automorphisms Fixing the Center Element-wise

TL;DR

This paper characterizes finite p-groups of nilpotency class 2 for which all central automorphisms fix the center element-wise, providing a complete set of necessary and sufficient conditions.

Contribution

It offers a new characterization of when central automorphisms act trivially on the center in finite p-groups of class 2, filling a gap in automorphism theory.

Findings

Identifies necessary and sufficient conditions for automorphisms to fix the center

Provides a classification for a specific class of p-groups

Enhances understanding of automorphism actions on group centers

Abstract

Let $G$ be a finite $p$-group of nilpotency class 2. We find necessary and sufficient conditions on $G$ such that each central automorphism of $G$ fixes the center of $G$ element-wise.