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

Abstract

A finite group is called ψ-divisible iff ψ(H)|ψ(G) for any subgroup H of a finite group G. Here, ψ(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 ψ-divisible groups still constitutes an open question. The aim of this paper is to make a connection between the ψ-divisibility property and graph theory. Hence, for a finite group G, we introduce a simple undirected graph called the ψ-divisibility graph of G. We denote it by ψ G . Its vertices are the non-trivial subgroups of G, while two distinct vertices H and K are adjacent iff H ⊂ K and ψ(H)|ψ(K) or K ⊂ H and ψ(K)|ψ(H). We prove that G is ψ-divisible iff ψ G has a universal (dominating) vertex. Also, we study various properties of ψ 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.