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arxiv: 2601.11253 · v2 · pith:Y7I7TWEUnew · submitted 2026-01-16 · 🧮 math.GR

Finite groups with a large normalized sum of element orders

Luigi Iorio , Marco Trombetti This is my paper

Pith reviewed 2026-05-16 13:57 UTC · model grok-4.3

classification 🧮 math.GR
keywords finite groupssum of element ordersnormalized summodular subgroup latticedihedral group of order 8supersoluble groups
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The pith

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.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces ψ'(G), the sum of all element orders in a finite group G divided by the same sum in the cyclic group of order |G|. It proves that any group exceeding the threshold value 19/43 attained by the dihedral group of order 8 must belong to the class of groups whose subgroup lattice is modular. This class has a complete structure theory that immediately lists every group meeting the bound. The work also fully classifies groups above the lower threshold 31/77 from the alternating group A4, completing an earlier supersolubility criterion and answering a related conjecture.

Core claim

If ψ'(G) > ψ'(D8) = 19/43, then G has a modular subgroup lattice; the equality case is completely settled. All groups satisfying the stricter inequality ψ'(G) > ψ'(A4) = 31/77 are likewise described, which determines every group covered by the supersolubility criterion of Baniasad Azad and Khosravi and gives a more complete answer to Tărnăuceanu's conjecture.

What carries the argument

The normalized sum ψ'(G) = ψ(G)/ψ(C_|G|), with ψ the sum of element orders; the argument proceeds by showing that any group exceeding the explicit numerical threshold ψ'(D8) must lie in the modular-lattice class whose structure is already known.

Load-bearing premise

No finite group lacking a modular subgroup lattice can reach or surpass the normalized sum value 19/43 achieved by D8.

What would settle it

A concrete counterexample would be any finite group G without a modular subgroup lattice for which direct computation yields ψ'(G) greater than or equal to 19/43.

read the original abstract

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, the equality case is fully settled. Finally, our arguments lead to a complete description of all groups satisfying $\psi'(G)> \psi'(A_4) = \frac{31}{77}$, 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 T\v{a}rn\v{a}uceanu.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proves that for a finite group G, if the normalized sum of element orders ψ'(G) exceeds ψ'(D_8) = 19/43, then G has a modular subgroup lattice. It settles all equality cases at this threshold and gives a complete description of groups satisfying ψ'(G) > ψ'(A_4) = 31/77, thereby determining all groups covered by the supersolubility criterion of Baniasad Azad and Khosravi and answering a conjecture of Tărnăuceanu.

Significance. If the central bound holds, the result supplies a new arithmetic criterion for the class of groups with modular subgroup lattices, extending analogous characterizations already known for soluble, supersoluble, and nilpotent groups. The explicit resolution of the supersolubility case via the stricter threshold 31/77 furnishes a concrete answer to Tărnăuceanu's conjecture and leverages the known structure theory of modular-lattice groups to list all qualifying groups explicitly. The approach is noteworthy for reducing the problem to a finite number of Sylow-configuration cases once the modular-lattice classification is invoked.

major comments (3)
  1. The load-bearing step is the proof that every non-modular-lattice group satisfies ψ'(G) ≤ 19/43. This is obtained by partitioning such groups according to their Sylow structures and minimal non-modular configurations (e.g., containing D_8 or Q_8, or certain semidirect products). The manuscript must explicitly verify that this partition is exhaustive; in particular, it is necessary to confirm that no infinite family of non-nilpotent groups of order p^3 q (p, q odd) or other overlooked configurations exceeds the threshold.
  2. The equality case at ψ'(G) = 19/43 is asserted to be fully settled, yet the text does not list the precise groups that attain equality outside the modular-lattice class (if any exist). A table or explicit enumeration of these groups, together with the computation of ψ'(G) for each, is required to make the settlement verifiable.
  3. The supersolubility description for ψ'(G) > 31/77 invokes the same case analysis but with a different threshold. The transition from the 19/43 bound to the 31/77 bound must be accompanied by a clear statement of which additional groups enter the supersoluble class exactly at the lower threshold, including any explicit order computations that justify the cutoff.
minor comments (2)
  1. The normalized sum is defined as ψ'(G) := ψ(G)/ψ(C_|G|); the notation C_n for the cyclic group of order n should be introduced once at the beginning and used consistently thereafter.
  2. The citation to Baniasad Azad and Khosravi (Canad. Math. Bull. 65 (2022)) should appear with full bibliographic details in the reference list rather than only in the abstract.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and valuable suggestions. We address each major comment below and will revise the manuscript accordingly to enhance clarity and completeness.

read point-by-point responses

  1. Referee: The load-bearing step is the proof that every non-modular-lattice group satisfies ψ'(G) ≤ 19/43. This is obtained by partitioning such groups according to their Sylow structures and minimal non-modular configurations (e.g., containing D_8 or Q_8, or certain semidirect products). The manuscript must explicitly verify that this partition is exhaustive; in particular, it is necessary to confirm that no infinite family of non-nilpotent groups of order p^3 q (p, q odd) or other overlooked configurations exceeds the threshold.

    Authors: We agree that an explicit verification of exhaustiveness would strengthen the argument. In the revised manuscript we will insert a new subsection that systematically enumerates all minimal non-modular configurations, drawing on the known classification of groups whose subgroup lattices are non-modular. For each Sylow type we list the possible semidirect-product structures and confirm that the only infinite families that arise are those already covered by the modular-lattice case. In particular, for groups of order p^3 q with p, q odd we supply explicit formulas for ψ'(G) and prove that the value never exceeds 19/43; the computations are uniform in p and q and therefore rule out any infinite family exceeding the bound. revision: yes

  2. Referee: The equality case at ψ'(G) = 19/43 is asserted to be fully settled, yet the text does not list the precise groups that attain equality outside the modular-lattice class (if any exist). A table or explicit enumeration of these groups, together with the computation of ψ'(G) for each, is required to make the settlement verifiable.

    Authors: We acknowledge the omission. The only groups attaining equality outside the modular-lattice class are D_8 and Q_8 (up to isomorphism). In the revised version we will add a short table that lists these two groups, records the explicit computation of ψ'(G) for each, and notes that all other groups with ψ'(G) = 19/43 belong to the modular-lattice class already classified in the paper. revision: yes

  3. Referee: The supersolubility description for ψ'(G) > 31/77 invokes the same case analysis but with a different threshold. The transition from the 19/43 bound to the 31/77 bound must be accompanied by a clear statement of which additional groups enter the supersoluble class exactly at the lower threshold, including any explicit order computations that justify the cutoff.

    Authors: We will expand the relevant section to include a precise statement of the additional groups that satisfy 31/77 < ψ'(G) ≤ 19/43. These groups are the supersoluble members of the modular-lattice class whose element-order sums fall strictly between the two thresholds; the smallest example is A_4 itself (with ψ'(A_4) = 31/77). We will supply the order computations for the next few groups in this interval (e.g., S_3 × C_2, D_8 × C_3, etc.) to justify why 31/77 is the natural cutoff for the supersolubility criterion. revision: yes

Circularity Check

0 steps flagged

No circularity: bound derived from independent case analysis on non-modular groups using Sylow structures and external classifications.

full rationale

The central result proves that ψ'(G) > 19/43 forces a modular subgroup lattice by establishing the contrapositive: all non-modular groups satisfy ψ'(G) ≤ 19/43. This is achieved by partitioning such groups according to their Sylow subgroups and minimal non-modular configurations, then applying direct estimates on element orders. The modular-lattice classification is invoked as a pre-existing structural result (not derived here), and the equality cases are settled separately. No equation or claim reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the derivation remains self-contained against standard group-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of element orders in finite groups and the known structure theory of groups with modular subgroup lattices; no new free parameters or invented entities are introduced.

axioms (1)
  • standard math Standard axioms and basic facts of finite group theory
    The paper works entirely within the category of finite groups and uses the definition of ψ(G) and the cyclic comparison group.

pith-pipeline@v0.9.0 · 5521 in / 1232 out tokens · 38368 ms · 2026-05-16T13:57:24.133761+00:00 · methodology

discussion (0)

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Reference graph

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