Finite groups with a large normalized sum of element orders

For a finite group G , let ψ (G) ψ ( G ) be the sum of the orders of its elements, and define the corresponding normalized sum as ψ '(G) :=ψ (G)/ψ (C|G|) ψ ′ ( G ) : = ψ ( G ) / ψ ( C | G | ) , where C|G| 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 '(D₈) = 19/43 ψ ′ ( G ) > ψ ′ ( D 8 ) = 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 '(A₄) = 31/77 ψ ′ ( G ) > ψ ′ ( A 4 ) = 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.