Year: 2026
Type: article
Abstract: For a finite group G , let $$\psi (G)$$ ψ ( G ) be the sum of the orders of its elements, and define the corresponding normalized sum as $$\psi '(G) :=\psi (G)/\psi (\mathcal {C}_{|G|})$$ ψ ′ ( G ) : = ψ ( G ) / ψ ( C | G | ) , where $$\mathcal {C}_{|G|}$$ 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 $$\psi '(G)>\psi '(D_8) = \frac{19}{43}$$ 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 $$\psi '(G)> \psi '(A_4) = \frac{31}{77}$$ 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.
Authors: Luigi Iorio, Marco Trombetti
Institution: University of Naples Federico II
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W7164943959