Finite groups with a large normalized sum of element orders
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866909038769340416 |
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| author | Iorio, Luigi Trombetti, Marco |
| author_facet | Iorio, Luigi Trombetti, Marco |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finite groups with a large normalized sum of element orders Iorio, Luigi Trombetti, Marco Group Theory 20D60, 20E34 (Primary) 20D30, 20F16 (Secondary) 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. |
| title | Finite groups with a large normalized sum of element orders |
| topic | Group Theory 20D60, 20E34 (Primary) 20D30, 20F16 (Secondary) |
| url | https://arxiv.org/abs/2601.11253 |