Finite groups with a large normalized sum of element orders
Luigi Iorio, Marco Trombetti
TL;DR
The paper addresses the problem of identifying finite groups with a large normalized sum of element orders ψ'(G). By leveraging sharp bounds derived from ψ(𝒞_n) and the modular-lattice structure of finite M-groups, the authors prove that any noncyclic G with ψ'(G)> rac{19}{43} must be an M-group and give an explicit, coprime-factorized classification into several families; they also completely determine the equality case ψ'(G)= rac{19}{43}. Extending the analysis, they describe all G with ψ'(G)> rac{31}{77} (the A_4 threshold), thereby resolving the supersolubility criterion conjecture in this range and relating results to known group-structure classifications. The work combines structural group theory (M-groups, P^*-groups, Iwasawa triples) with ψ-function techniques to produce a self-contained, constructive classification that avoids heavy computational enumeration. Overall, the paper clarifies how large ψ'(G) values constrain group structure and settles key conjectures about supersolubility and modular lattices in finite groups.
Abstract
For a finite group $G$, let $ψ(G)$ be the sum of the orders of its elements, and define the corresponding normalized sum as $ψ'(G) := ψ(G)/ψ(\mathcal{C}_{|G|})$, where $\mathcal{C}_{|G|}$ is the cyclic group of the same order as $G$. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our main result establishes that if $ψ'(G)>ψ'(D_8) = \frac{19}{43}$, then $G$ belongs to the well-understood class of groups with a modular subgroup lattice, whose structure theory allows us to readily identify all groups satisfying this bound. Moreover, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying $ψ'(G)> ψ'(A_4) = \frac{31}{77}$, thereby fully determining the groups covered by the supersolubility criterion of Baniasad Azad and Khosravi [Canad. Math. Bull. 65 (2022), 30--38], and thus providing a more complete answer to a corresponding conjecture of Tǎrnǎuceanu.