Identifying structural characteristics of finite groups by the sum of their element orders

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. In this paper, we study the function [Formula: see text], where [Formula: see text] is a real number, and we investigate the influence of [Formula: see text] on cyclicity, nilpotency, supersolvability and solvability of a finite group [Formula: see text].