On groups whose conjugacy class sizes are not divisible by each other

TL;DR

This paper investigates finite groups whose conjugate graph, based on conjugacy class sizes, consists solely of isolated points, revealing structural properties related to divisibility among class sizes.

Contribution

It introduces the conjugate graph concept for finite groups and characterizes groups with graphs that are just isolated points, advancing understanding of class size divisibility.

Findings

Conjugate graph is a set of points for certain finite groups.

Characterization of groups with trivial conjugate graphs.

Insights into divisibility relations among conjugacy class sizes.

Abstract

Let $G$ be a finite group and $N(G)$ be the set of its conjugacy class sizes excluding~$1$. Let us define a directed graph $\Gamma(G)$, the set of vertices of this graph is $N(G)$ and the vertices $x$ and $y$ are connected by a directed edge from $x$ to $y$ if $x$ divides $y$ and $N(G)$ does not contain a number $z$ different from $x$ and $y$ such that $x$ divides $z$ and $z$ divides $y$. We will call the graph $\Gamma(G)$ the conjugate graph of the group $G$. In this work, we will study finite groups whose conjugate graph is a set of points.