Sums of element orders in groups of odd order

Abstract

For a finite group [Formula: see text], let [Formula: see text] denote the sum of element orders of [Formula: see text]. If [Formula: see text] is a positive integer let [Formula: see text] be the cyclic group of order [Formula: see text]. It is known that [Formula: see text] is the maximum value of [Formula: see text] on the set of groups of order [Formula: see text], and [Formula: see text] if and only if [Formula: see text] is cyclic of order [Formula: see text]. In this paper, we investigate the second largest value of [Formula: see text] on the set of groups of order [Formula: see text] and the structure of groups [Formula: see text] of order [Formula: see text] with this value of [Formula: see text] when [Formula: see text] is odd.