pith. sign in

arxiv: 2511.08320 · v2 · submitted 2025-11-11 · 🧮 math.GR

On the Sum of Element Orders in Finite Abelian Groups

Mohsen Amiri This is my paper

Pith reviewed 2026-05-17 23:41 UTC · model grok-4.3

classification 🧮 math.GR
keywords sum of element ordersfinite abelian groupsLCM-groupsorder typegroup isomorphismelement orders
0
0 comments X

The pith

Finite LCM-groups of the same order have equal sums of element orders precisely when they share the same order type.

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

The paper proves that among finite LCM-groups of a fixed order, the sum of element orders ψ(G) equals ψ(H) if and only if G and H have the same order type. This strengthens an earlier result that held only for abelian p-groups and settles a conjecture asserting the same uniqueness for all finite abelian groups. A reader would care because ψ(G) then functions as a practical classifier that recovers the group's structural invariants without enumerating all isomorphism types. The result applies directly to the abelian case since finite abelian groups form a subclass of LCM-groups.

Core claim

For finite LCM-groups G and H of the same order, ψ(G) = ψ(H) if and only if G and H have the same order type, where ψ(G) denotes the sum over all g in G of the order o(g) of g.

What carries the argument

The function ψ(G) = sum of o(g) over all elements g, which the paper shows separates distinct order types inside the class of LCM-groups of fixed order.

If this is right

  • Finite abelian p-groups of equal order are isomorphic exactly when their element-order sums coincide.
  • Tărnăuceanu's conjecture holds for every finite abelian group.
  • Order type becomes recoverable from the single numerical value ψ(G) whenever the group is an LCM-group.
  • Any two LCM-groups with the same ψ value and same order must share all invariants encoded by order type.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same numerical invariant might separate isomorphism classes inside larger families that contain the LCM-groups.
  • Computing ψ(G) could serve as a fast preliminary filter before running full isomorphism tests on groups of moderate order.
  • Similar sum-of-orders functions might distinguish other algebraic structures whose invariants are encoded by partitions or type data.

Load-bearing premise

The groups in question must belong to the class of LCM-groups and the notion of order type must capture exactly the data needed to distinguish their structures.

What would settle it

Two LCM-groups of identical order but different order types that nevertheless satisfy ψ(G) = ψ(H) would disprove the stated equivalence.

read the original abstract

Let $\psi(G) = \sum_{g \in G} o(g)$ denote the sum of element orders of a finite group $G$. It is known that among groups of order $n$, the cyclic group $C_n$ maximizes $\psi$. T\u{a}rn\u{a}uceanu proved that two finite abelian $p$-groups of the same order are isomorphic if and only if they have the same sum of element orders, and conjectured this for arbitrary finite abelian groups. In this paper, we confirm the conjecture by proving a stronger result: for finite $LCM$-groups $G$ and $H$ of the same order, $\psi(G) = \psi(H)$ if and only if $G$ and $H$ have the same order type.

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

2 major / 1 minor

Summary. The manuscript proves that for finite LCM-groups G and H of the same order, ψ(G) = ψ(H) if and only if G and H have the same order type. This equivalence is presented as confirming Tărnăuceanu's conjecture that two finite abelian groups of the same order are isomorphic precisely when they have the same sum of element orders.

Significance. If the central equivalence holds and the class of LCM-groups contains all finite abelian groups, the result strengthens the known p-group case by providing an if-and-only-if characterization via ψ and order type. The direct proof (rather than a reduction to a fitted model) is a positive feature.

major comments (2)
  1. [Introduction / statement of main result] The confirmation of Tărnăuceanu's conjecture for arbitrary finite abelian groups rests on the unverified claim that every finite abelian group is an LCM-group. No explicit lemma establishing this inclusion (or showing that the LCM property holds for groups with multiple prime factors) appears in the provided text; without it the stronger statement for LCM-groups does not entail the full conjecture.
  2. [§2 (definitions and preliminary lemmas)] The definitions of 'LCM-group' and 'order type' are load-bearing for the equivalence but are not fully visible in the abstract or surrounding discussion; the proof of the if-and-only-if direction must be checked for any post-hoc choices in these definitions.
minor comments (1)
  1. [Notation and preliminaries] Notation for ψ(G) and the order type should be introduced once and used consistently; any intermediate lemmas on the sum of orders should be numbered for easy reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to make the connection to Tărnăuceanu's conjecture fully rigorous and to improve the visibility of key definitions. We address each major comment below and have incorporated revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Introduction / statement of main result] The confirmation of Tărnăuceanu's conjecture for arbitrary finite abelian groups rests on the unverified claim that every finite abelian group is an LCM-group. No explicit lemma establishing this inclusion (or showing that the LCM property holds for groups with multiple prime factors) appears in the provided text; without it the stronger statement for LCM-groups does not entail the full conjecture.

    Authors: We agree that an explicit verification is required for the claim to entail the full conjecture. The definition of an LCM-group is formulated so that the property holds for all finite abelian groups (via the fundamental theorem), but we acknowledge that this inclusion was not stated as a separate lemma. In the revised manuscript we have added Lemma 2.3 in Section 2, which proves that every finite abelian group is an LCM-group by verifying the relevant LCM condition on element orders for both p-groups and groups of composite order. revision: yes

  2. Referee: [§2 (definitions and preliminary lemmas)] The definitions of 'LCM-group' and 'order type' are load-bearing for the equivalence but are not fully visible in the abstract or surrounding discussion; the proof of the if-and-only-if direction must be checked for any post-hoc choices in these definitions.

    Authors: The formal definitions appear at the beginning of Section 2, immediately before the preliminary results, and the proof of the main theorem uses precisely those definitions. No post-hoc adjustments were made. To improve accessibility we have inserted a brief paragraph at the end of the introduction that recalls the definitions of LCM-group and order type verbatim. The if-and-only-if argument was developed with these definitions fixed in advance. revision: yes

Circularity Check

0 steps flagged

No circularity: direct proof for LCM-groups stands independently

full rationale

The paper states a theorem for finite LCM-groups and notes that this confirms Tărnăuceanu's conjecture once the class inclusion is granted. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The derivation is presented as an explicit proof rather than a renaming or ansatz imported from prior work by the same authors. The class-inclusion question raised in the skeptic note is a completeness issue, not a circular reduction of the stated result to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the standard structure theorem for finite abelian groups, the definition of element order, and whatever additional properties define an LCM-group; these are standard or paper-specific but not quantified as free parameters.

axioms (1)
  • standard math Finite abelian groups decompose as direct sums of cyclic groups of prime-power order (fundamental theorem of finite abelian groups).
    Implicitly used when discussing order type and isomorphism classes.

pith-pipeline@v0.9.0 · 5423 in / 1158 out tokens · 32584 ms · 2026-05-17T23:41:06.120237+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

  1. [1]

    Amiri and I

    M. Amiri and I. Lima (2022) The order of the product of two elements in the periodic groups, Communications in Algebra, 50:8, 3473-3480

    work page 2022

  2. [2]

    Amiri, S.M

    H. Amiri, S.M. Jafarian Amiri, I.M. Isaacs, Sums of element orders in finite groups, Comm. Algebra 37 (9) 2978-2980(2009)

    work page 2009

  3. [3]

    Jafarian Amiri, M

    S.M. Jafarian Amiri, M. Amiri, Second maximum sum of element orders on finite groups, J. Pure Appl. Algebra 218 (3) (2014), 531-539

    work page 2014

  4. [4]

    Jafarian Amiri, M

    S.M. Jafarian Amiri, M. Amiri, Sum of the Products of the Orders of Two Distinct Elements in Finite Groups, Communication in Algebra, v. 42, p. 5319-5328, 2014

    work page 2014

  5. [5]

    Amiri, On a bijection between a finite group and cyclic group, Journal of Pure and Applied Algebra,Volume 228, Issue 7, July 2024, 107632

    M. Amiri, On a bijection between a finite group and cyclic group, Journal of Pure and Applied Algebra,Volume 228, Issue 7, July 2024, 107632

    work page 2024

  6. [6]

    Baniasad Asad, B

    M. Baniasad Asad, B. Khosravi, A Criterion for Solvability of a Finite Group by the Sum of Element Orders, J. Algebra 516 (2018), 115-124

    work page 2018

  7. [7]

    Chew, A.Y.M

    C.Y. Chew, A.Y.M. Chin, C.S. Lim A recursive formula for the sum of element orders of finite Abelian groups Results Math., 72 (2017), pp. 1897-1905

    work page 2017

  8. [8]

    Jafarian Amiri Second maximum sum of element orders on finite nilpotent groups Commun

    S.M. Jafarian Amiri Second maximum sum of element orders on finite nilpotent groups Commun. Algebra, 41 (2013), pp. 2055-2059

    work page 2013

  9. [9]

    Jafarian Amiri, M

    S.M. Jafarian Amiri, M. Amiri, Sum of the products of the orders of two distinct elements in finite groups, Comm. Algebra 42 (12) (2014), 5319-5328

    work page 2014

  10. [10]

    Jafarian Amiri, M

    S.M. Jafarian Amiri, M. Amiri, Second maximum sum of element orders on finite groups, J. Pure Appl. Algebra 218 (2014) 531–539

    work page 2014

  11. [11]

    Herzog, P

    M. Herzog, P. Longobardi, M. Maj, Two new criteria for solvability of finite groups in finite groups, J. Algebra 511 (2018), 215-226

    work page 2018

  12. [12]

    Herzog, P

    M. Herzog, P. Longobardi, M. Maj Sums of element orders in groups of order 2m with m odd Commun. Algebra, 47 (2019), pp. 2035-2948

    work page 2019

  13. [13]

    Herzog, P

    M. Herzog, P. Longobardi, M. Maj The second maximal groups with respect to the sum of element orders J. Pure Appl. Algebra, 225 (2021), Article 106531

    work page 2021

  14. [14]

    Martin Isaacs, Finite Group Theory, American Mathematical Soc., Vol

    I. Martin Isaacs, Finite Group Theory, American Mathematical Soc., Vol. 92 (2008)

    work page 2008

  15. [15]

    Kishore Dey and A

    H. Kishore Dey and A. Mondal, An exact upper bound for the sum of powers of element orders in non-cyclic finite groups, Journal of Pure and Applied Algebra Volume 228, Issue 6, June 2024, 107580

    work page 2024

  16. [16]

    Marefat, A

    Y. Marefat, A. Iranmanesh, A. Tehranian, On the sum of element orders of finite simple groups, J. Algebra Appl. 12 (7) (2013), 135-138

    work page 2013

  17. [17]

    V. D. Mazurov, E. I. Khukhro, The Kourovka Notebook. Unsolved Problems in Group Theory, 18th ed., Institute of Mathematics, Russian Academy of Sciences, Siberrian Division, Novosibirsk, arXiv:1401.0300v3 [math. GR] (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  18. [18]

    R. Shen, G. Chen, C. Wu On groups with the second largest value of the sum of element orders Commun. Algebra, 43 (2015), pp. 2618-2631. ON THE SUM OF ELEMENT ORDERS. . . 21

    work page 2015

  19. [19]

    T˘ arn˘ auceanu, Detecting structural properties of finite groups by the sum of element orders Isr

    M. T˘ arn˘ auceanu, Detecting structural properties of finite groups by the sum of element orders Isr. J. Math., 238 (2020), pp. 629-637

    work page 2020

  20. [20]

    T˘ arn˘ auceanu,Finite groups determined by an inequality of the order of their ele- ments, Publ

    M. T˘ arn˘ auceanu,Finite groups determined by an inequality of the order of their ele- ments, Publ. Math. Debrecen,80(2012), no. 3-4, 457-463

    work page 2012

  21. [21]

    Al. I. Cuza

    M. T˘ arn˘ auceanu,On the sum of element orders of finite abelian groups, An. S ¸tiint ¸. Univ. “Al. I. Cuza” Ia¸ si. Mat. (N.S.)60(2014), no. 1, 113–122. DOI: 10.2478/aicu- 2013-0013. Faculdade de Matem´atica, Universidade Federal de Uberl ˆandia, Av. J. N. ´Avila 2121, 38408-902 Uberl ˆandia - MG, Brazil

    work page doi:10.2478/aicu- 2014