A graph related to the sum of element orders of a finite group

TL;DR

This paper introduces a graph-theoretic approach to studying $ ext{psi}$-divisible finite groups, linking group properties to the structure of a new graph called the $ ext{psi}$-divisibility graph, and explores its properties especially for cyclic groups.

Contribution

It establishes a novel connection between $ ext{psi}$-divisibility in finite groups and graph theory, defining the $ ext{psi}$-divisibility graph and characterizing $ ext{psi}$-divisible groups via graph properties.

Findings

A finite group is $ ext{psi}$-divisible iff its $ ext{psi}$-divisibility graph has a universal vertex.

The paper characterizes properties of the $ ext{psi}$-divisibility graph for cyclic groups.

It provides insights into the structure of $ ext{psi}$-divisible groups through graph analysis.

Abstract

A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of square-free order. The existence of non-abelian $\psi$-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the $\psi$-divisibility property and graph theory. Hence, for a finite group $G$, we introduce a simple undirected graph called the $\psi$-divisibility graph of $G$. We denote it by $\psi_G$. Its vertices are the non-trivial subgroups of $G$, while two distinct vertices $H$ and $K$ are adjacent iff $H\subset K$ and $\psi(H)|\psi(K)$ or $K\subset H$ and $\psi(K)|\psi(H)$. We prove that $G$ is $\psi$-divisible iff $\psi_G$ has a universal (dominating) vertex. Also, we study various properties of $\psi_G$, when $G$ is a finite cyclic group. The choice of restricting our study to this specific class of groups is motivated in the paper.