On a divisibility condition related to the sum of element orders of a finite group

Abstract

Given a finite group G, we denote by [Formula: see text](G) the sum of element orders of G. In this note, we determine finite groups G such that [Formula: see text]H[Formula: see text] – [Formula: see text]K[Formula: see text] divides [Formula: see text](H) – [Formula: see text](K) for all subgroups K ≤ H ≤ G. Two weaker conditions are also proposed.