Finite groups with a large normalized sum of element orders

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.