On divisibility graph for simple Zassenhaus groups

TL;DR

This paper investigates the structure of the divisibility graph for simple Zassenhaus groups, revealing specific properties and configurations of conjugacy class sizes within these groups.

Contribution

It provides a detailed analysis of the divisibility graph for simple Zassenhaus groups, a topic not previously explored in depth.

Findings

Characterization of the divisibility graph for simple Zassenhaus groups

Identification of adjacency patterns based on divisibility of conjugacy class sizes

Insights into the structure of conjugacy classes in these groups

Abstract

The divisibility graph $D(G)$ for a finite group $G$ is a graph with vertex set $cs~(G)\setminus\{1\}$ where $cs~(G)$ is the set of conjugacy class sizes of $G$. Two vertices $a$ and $b$ are adjacent whenever $a$ divides $b$ or $b$ divides $a$. In this paper we will find $D(G)$ where $G$ is a simple Zassenhaus group.