# Autocommuting probability of a finite group

## TL;DR

This paper investigates the autocommuting probability in finite groups, providing formulas, bounds, and characterizations, and shows its invariance under autoisoclinism, enriching the understanding of automorphism actions.

## Contribution

It introduces a generalized autocommuting probability for finite groups, derives formulas and bounds, and proves invariance under autoisoclinism, advancing group automorphism theory.

## Key findings

- Derived a computing formula for autocommuting probability
- Established bounds and characterizations of finite groups based on this probability
- Proved invariance of the generalized autocommuting probability under autoisoclinism

## Abstract

Let $G$ be a finite group and $\Aut(G)$ the automorphism group of $G$. The autocommuting probability of $G$, denoted by $\Pr(G, \Aut(G))$, is the probability that a randomly chosen automorphism of $G$ fixes a randomly chosen element of $G$. In this paper, we study $\Pr(G, \Aut(G))$ through a generalization. We obtain a computing formula, several bounds and characterizations of $G$ through $\Pr(G, \Aut(G))$. We conclude the paper by showing that the generalized autocommuting probability of $G$ remains unchanged under autoisoclinism.