Finite groups with a large normalized sum of element orders

TL;DR

This paper characterizes finite groups with a large normalized sum of element orders, establishing bounds that identify groups with specific structural properties and fully describing those exceeding certain thresholds.

Contribution

It provides new criteria based on the normalized sum of element orders to classify groups with modular subgroup lattices and confirms conjectures related to supersolubility.

Findings

Groups with normalized sum > 19/43 belong to groups with modular subgroup lattices.

The exact equality case for the 19/43 bound is fully characterized.

Complete description of groups with normalized sum > 31/77, confirming related conjectures.

Abstract

For a finite group $G$, let $\psi(G)$ be the sum of the orders of its elements, and define the corresponding normalized sum as $\psi'(G) := \psi(G)/\psi(\mathcal{C}_{|G|})$, 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 $\psi'(G)>\psi'(D_8) = \frac{19}{43}$, 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 $\psi'(G)> \psi'(A_4) = \frac{31}{77}$, 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 T\v{a}rn\v{a}uceanu.