{"paper":{"title":"Finite groups with a large normalized sum of element orders","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"If the normalized sum of element orders in a finite group exceeds 19/43, the value for the dihedral group of order 8, then the group has a modular subgroup lattice.","cross_cats":[],"primary_cat":"math.GR","authors_text":"Luigi Iorio, Marco Trombetti","submitted_at":"2026-01-16T13:01:49Z","abstract_excerpt":"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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if 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","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The critical threshold is exactly ψ'(D8), which is assumed to be the largest value attained by groups outside the modular-lattice class; this relies on exhaustive case analysis or classification results for small-order groups that are not detailed in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Finite groups with ψ'(G) > 19/43 have modular subgroup lattices; all groups with ψ'(G) > 31/77 are classified, completing the supersolubility criterion.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If the normalized sum of element orders in a finite group exceeds 19/43, the value for the dihedral group of order 8, then the group has a modular subgroup lattice.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"5130e7e6d96d8f34c9dfc68164de3b982d851f3296a3d305f901c8de3af974cf"},"source":{"id":"2601.11253","kind":"arxiv","version":2},"verdict":{"id":"063f679e-4281-42c1-af78-691f701123d2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T13:57:24.133761Z","strongest_claim":"if 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","one_line_summary":"Finite groups with ψ'(G) > 19/43 have modular subgroup lattices; all groups with ψ'(G) > 31/77 are classified, completing the supersolubility criterion.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The critical threshold is exactly ψ'(D8), which is assumed to be the largest value attained by groups outside the modular-lattice class; this relies on exhaustive case analysis or classification results for small-order groups that are not detailed in the abstract.","pith_extraction_headline":"If the normalized sum of element orders in a finite group exceeds 19/43, the value for the dihedral group of order 8, then the group has a modular subgroup lattice."},"references":{"count":13,"sample":[{"doi":"","year":2009,"title":"Sums of element orders in finite groups","work_id":"83c5e964-c3e4-49d7-a3f0-06a67b4250f4","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"A criterion for solvability of a finite group by the sum of element orders","work_id":"ad4d6d96-86c4-48bf-a267-28f04a688360","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2022,"title":"On two conjectures about the sum of element orders","work_id":"b00ba08f-fdce-4d46-8c27-6cec432d2bd2","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"A result on the sum of element orders of a finite group","work_id":"13c2cb74-12ed-4ba6-96c9-a0471b97ca6a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"GAP – Groups, Algorithms, and Programming","work_id":"f5ac4121-fdcb-4330-8b22-0519f5eb31fc","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"ddb6ca392b1e47d38a922f009c1c9b03ead7526f339c3aa818499fc028d9c025","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}