# Commuting graph of $A$-orbits

## TL;DR

This paper introduces the commuting graph of a finite group action and investigates how its structure influences the properties of the group, providing new insights into the interplay between group actions and graph theory.

## Contribution

It defines the commuting graph for a group action and explores structural implications under specific graph-theoretic conditions, a novel approach in the study of group automorphisms.

## Key findings

- Characterization of group structure via commuting graph properties
- Conditions under which the commuting graph reveals group automorphism features
- New connections between graph theory and automorphism group analysis

## Abstract

Let $A$ be a finite group acting by automorphisms on the finite group $G$. We introduce the commuting graph $\Gamma (G,A)$ of this action and study some questions related to the structure of $G$ under certain graph theoretical conditions on $\Gamma (G,A)$.