On a divisibility condition related to the sum of element orders of a finite group

Marius T\u{a}rn\u{a}uceanu

On a divisibility condition related to the sum of element orders of a finite group

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Pith's one-line read A subgroup divisibility condition characterizes elementary abelian p-groups.

desk verdict A short, clean note proving that the subgroup-pair divisibility condition on ψ characterizes precisely the p-groups of exponent p; the proof is elementary and correct, with only minor expository caveats. read the letter →

arxiv 2608.10036 v1 pith:GNOJ4ZMG submitted 2026-08-10 math.GR

Marius Tărnăuceanu This is my paper · ORCID

classification math.GR MSC 20D6020D15

keywords sumofelementordersdivisibilityconditionselementaryabelianp-groupsCP1-groupsFrobeniusgroupsalternatinggroupA5finiteclassification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks when the arithmetic condition $|H|-|K|$ divides $\psi(H)-\psi(K)$ for every nested pair of subgroups $K\leq H\leq G$ forces a recognizable group structure, where $\psi(G)$ is the sum of the orders of the group's elements. Its main theorem answers: a finite group satisfies this condition exactly when it is a $p$-group of exponent $p$, meaning every nonidentity element has order $p$ and the group is a direct product of copies of the cyclic group $C_p$. The proof shows the condition rules out elements of order $p^2$ and elements of order $pq$, so the group must be one whose nonidentity elements all have prime order; the classification of such groups then leaves only the elementary abelian case as a survivor. The paper also records two weaker divisibility conditions and gives examples showing the hierarchy is nontrivial, for instance $\mathbb{Z}_6$ satisfies the single-subgroup version without satisfying the full condition.

What carries the argument

The central object is the function $\psi(G)=\sum_{x\in G} o(x)$, the sum of the orders of all elements of $G$, together with the subgroup-divisibility relation it is tested against. The load-bearing tool is Lemma 2.1, the classification of CP1-groups — finite groups in which every nonidentity element has prime order — which says such a group is either a $p$-group of exponent $p$, a Frobenius group with an elementary abelian $p$-kernel and a complement of prime order $q$, or the alternating group $A_5$. This classification carries the necessity argument: after divisibility forces $G$ to be CP1, the classification reduces the problem to three cases, and two of them are eliminated by counting elements of order $q$ and by a direct calculation in $A_5$. The function $\psi$ is computed on cyclic subgroups to rule out elements of order $p^2$ and $pq$, and on the whole group in the Frobenius and $A_5$ cases.

What would settle it

Run an exhaustive computation of $\psi$ on every subgroup pair for all finite groups of order up to 60 that are not elementary abelian $p$-groups; the theorem predicts a divisibility failure in each one, with explicit witnesses supplied by the proof — for $A_5$ and a subgroup of order 2, for instance, $58\nmid 208$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: for a finite group $G$, the condition $$|H|-|K|\ \big|\ \psi(H)-\psi(K)\quad\text{for all subgroups }K\leq H\leq G$$ holds if and only if $G$ is a $p$-group of exponent $p$. The proof first shows that the condition excludes elements of order $p^2$ and of order $pq$, so every nonidentity element has prime order; quoting the CP1-group classification, the only remaining candidates are the elementary abelian $p$-groups, the Frobenius groups with elementary abelian kernel and prime-order complement, and the alternating group $A_5$. Counting element orders in the Frobenius case and checking $A_5$ against a subgroup of order $2$ produce divisibility contradictions, so the elementary abelian $p$-groups are the only survivors. Conversely, in any elementary abelian $p$-group, $\psi(H)=p^{\alpha+1}-p+1$ when $|H|=p^\alpha$, and the divisibility follows from $p^\alpha-p^\beta \mid p^{\alpha+1}-p^{\beta+1}$.

Load-bearing premise

The argument depends on accepting the classification result that a finite group whose nonidentity elements all have prime order is either an elementary abelian $p$-group, a Frobenius group with an elementary abelian kernel and prime-order complement, or the alternating group $A_5$; if the classification missed a case, the 'only if' direction would not go through.

Editorial extensions

If this is right

Every finite group satisfying the full divisibility condition is elementary abelian, so the condition is a complete structural fingerprint of direct products of copies of $C_p$.
The theorem provides explicit witness pairs for failure: an element of order $p^2$ fails at $\langle a\rangle\geq 1$, an element of order $pq$ fails at $\langle b\rangle\geq \langle b^q\rangle$, and $A_5$ fails at a subgroup of order $2$.
Any group satisfying the full condition is a CP1-group, so the condition is strictly stronger than having all nonidentity elements of prime order.
The weaker condition $|H|-1\mid\psi(H)-1$ for all $H$ does not force prime-power order, since $\mathbb{Z}_6$ satisfies it; the still weaker condition (3) is satisfied by $C_8$ and the dicyclic group $Dic_3$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An extension not pursued in the paper: it may be enough to impose the divisibility on far fewer subgroup pairs, such as the pairs along a chief series; a concrete test would be whether requiring divisibility only for chief factors still forces an elementary abelian structure.
The line of examples $\mathbb{Z}_6$, $C_8$, and $Dic_3$ suggests a hierarchy of divisibility conditions of increasing strength; one could quantify how much of the finite-group landscape each level cuts out as a measure of how evenly element orders are distributed across subgroups.
Because $\psi$ is determined by the multiset of element orders, the theorem can be read as saying that a very strong arithmetic regularity of that multiset across all subgroups leaves no room for nonabelian or mixed-prime structure; a natural stress test would be the Frobenius groups with non-elementary-abelian kernels, where the proof's count of elements of order $q$ would break down.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This is a short, clean note. It proves that a finite group G satisfies

|H| − |K| divides ψ(H) − ψ(K) for all subgroups K ≤ H ≤ G

if and only if G is a p-group of exponent p. That is a tidy result, and the proof is correct as far as I can tell.

What is actually new: the condition over all subgroup pairs is new, and the characterization is neat. The proof reduces quickly to the CP1 classification, and the two non-p-group cases (Frobenius and A5) are excluded by straightforward arithmetic. The sufficiency direction is one line: for exponent-p p-groups, ψ(H) = 1 + p(|H| − 1), so the difference is p(|H| − |K|). I checked the divisibility calculations and the Frobenius element count; they all work.

The soft spots are minor. The statement as written does not handle K = H, where both sides are zero; you need the convention 0|0 or an implicit restriction to K < H. That is an expository fix, not a mathematical one. Section 3 is thin: it proposes two weaker conditions and gives examples, but stops there. That is fine for a note, but do not expect a theory. Also, the reliance on the full CP1 classification is a bit heavy for the result, but it is standard and used carefully.

One thing to keep straight: the theorem says p-groups of exponent p, which includes nonabelian groups for odd p (e.g., the extraspecial groups of order p^3 and exponent p). It does not characterize only elementary abelian groups, so do not let a summary mislead you.

Who benefits: people working on the ψ function and related divisibility conditions. It is a nice addition to that line, not a breakthrough. I would send it to peer review; a good referee will catch the K = H convention and maybe ask for a remark on the nonabelian examples. I would not cite it myself in the next year, but I would point a student to it as a clean exercise in using the CP1 classification.

Recommendation: accept after minor revisions.

Referee Report

0 major / 5 minor

Summary. The paper studies finite groups G for which |H|-|K| divides ψ(H)-ψ(K) for all subgroups K≤H≤G, where ψ is the sum of element orders. Theorem 1.1 states that these are precisely the finite p-groups of exponent p. The proof first excludes elements of order p^2 and pq, thereby showing G is a CP1-group; using the classification of CP1-groups it rules out the Frobenius and A5 cases by direct divisibility checks, and then verifies the condition for p-groups of exponent p. Section 3 proposes two weaker divisibility conditions, with examples showing the hierarchy is strict.

Significance. The main result is a clean, natural characterization and appears to be correct. The proof is short and elementary apart from the standard appeal to the classification of CP1-groups; all divisibility computations in Section 2 check out. The paper is honest about what it proves and does not overclaim: the conclusion is 'p-group of exponent p', which correctly includes nonabelian examples for odd p. This is a useful contribution to the literature on sums of element orders and should interest specialists in the area.

minor comments (5)

[Section 2, proof of Theorem 1.1] The inference 'It follows that G is a CP1-group' is correct but implicit: one should state that every composite integer has a divisor of the form p^2 or pq, so an element of composite order has a power of order p^2 or pq.
[Section 2, first and second paragraphs] The two 'contradiction' claims in the exclusions of orders p^2 and pq are not fully shown; adding the modular reductions (e.g. p(p^2+1)≡-2 mod p+1, and gcd(p,q)=1) would make the argument easier to follow.
[Theorem 1.1 / condition (1)] The statement should clarify the divisibility convention when |H|=|K| (in particular 0|0, or restrict (1) to K<H), and likewise for the trivial group.
[Section 2, Frobenius case] The paragraph counting (q-1)p^n elements of order q uses the standard fact that distinct Frobenius complements intersect trivially; a brief justification or citation would improve completeness.
[Section 3] The examples for conditions (2) and (3) are asserted without verification; since they are used to show the conditions are strictly weaker, one or two sentences of verification would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof uses only direct arithmetic and an independent external classification theorem.

full rationale

The paper derives Theorem 1.1 from the hypothesis by elementary arguments, without any fitted parameter, prediction, or self-referential premise. In the necessity direction, the author excludes elements of order p^2 and pq by direct divisibility contradictions, concluding that G has only elements of prime order (a CP1-group). The subsequent use of Lemma 2.1, the classification of CP1-groups, is a citation to independent prior work by Deaconescu and by Cheng, Deaconescu, Lang and Shi; it is not authored by Tarnauceanu, not derived in this paper, and not equivalent to the theorem being proved. The two remaining cases (Frobenius groups and A5) are eliminated by straightforward counting and divisibility arguments, with no circular dependence on the claimed characterization. In the sufficiency direction, the paper directly computes that for a p-group of exponent p, every subgroup H satisfies ψ(H)=1+p(|H|−1), so for K≤H, |H|−|K| divides ψ(H)−ψ(K)=p(|H|−|K|). Thus the central claim is a genuine theorem proven from stated hypotheses and independent literature, rather than a restatement of its assumptions. The weaker conditions in Section 3 are exploratory and do not affect the main result. No circularity is present; the score reflects a completely self-contained argument apart from standard external theorems, which are legitimate evidence.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

No free parameters, no invented entities. The proof depends on two external, established results: the CP1-group classification (Lemma 2.1) and standard Frobenius group counting facts. These are reasonable and properly cited, so the ledger burden is modest.

assumptions (2)

domain assumption Classification of CP1-groups (Lemma 2.1): every finite group with all nonidentity elements of prime order is either nilpotent and a p-group of exponent p, a Frobenius group with kernel a p-group of exponent p and cyclic complement of prime order q, or isomorphic to A5.
Invoked in the proof of Theorem 1.1 to reduce G to three cases after proving G is CP1; if this classification is incorrect or incomplete, the necessity direction fails.
domain assumption Frobenius group element counting: in the Frobenius group with kernel P of order p^n and complement Q of order q, the nonidentity elements of order q are exactly the nonidentity elements of the p^n conjugates of Q, and distinct conjugates intersect trivially.
Used in the same paragraph to compute ψ(G). These are standard Frobenius group facts, but not proved in the note.

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Pith. "Pith review of On a divisibility condition related to the sum of element orders of a finite group." pith.science (2026). https://pith.science/paper/GNOJ4ZMG

@misc{pith2026260810036,
  author       = {Pith},
  title        = {Pith review of: On a divisibility condition related to the sum of element orders of a finite group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GNOJ4ZMG}},
  note         = {Machine review of arXiv:2608.10036}
}

abstract

Given a finite group $G$, we denote by $\psi(G)$ the sum of element orders of $G$. In this note, we determine finite groups $G$ such that $|H|-|K|$ divides $\psi(H)-\psi(K)$ for all subgroups $K\leq H\leq G$. Two weaker conditions are also proposed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 10 canonical work pages

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2014 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Amiri, S.M

H. Amiri, S.M. Jafarian Amiri and I.M. Isaacs,Sums of element orders in finite groups, Comm. Algebra37(2009), 2978-2980. 4

work page 2009

[2] [2]

Amiri and S.M

M. Amiri and S.M. Jafarian Amiri,Characterization of finite groups by a bijection with a divisible property on the element orders, Comm. Algebra 45(2017), 3396-3401

work page 2017

[3] [3]

Amiri,On a bijection between a finite group and cyclic group, J

M. Amiri,On a bijection between a finite group and cyclic group, J. Pure Appl. Algebra228(2024), article ID 107632

work page 2024

[4] [4]

Cheng, M

K.N. Cheng, M. Deaconescu, L.M. Lang and W. Shi,Corrigendum and addendum to ”Classification of finite groups with all elements of prime order”, Proc. Amer. Math. Soc.117(1993), 1205-1207

work page 1993

[5] [5]

Deaconescu,Classification of finite groups with all elements of prime order, Proc

M. Deaconescu,Classification of finite groups with all elements of prime order, Proc. Amer. Math. Soc.106(1989), 625-629

work page 1989

[6] [6]

Harrington, L

J. Harrington, L. Jones and A. Lamarche,Characterizing finite groups using the sum of the orders of the elements, Int. J. Comb. (2014), article ID 835125

work page 2014

[7] [7]

Herzog, P

M. Herzog, P. Longobardi and M. Maj,New criteria for solvability, nilpotency and other properties of finite groups in terms of the order elements or subgroups, Int. J. Group Theory12(2023), 35-44

2023

[8] [8]

Isaacs,Finite group theory, Amer

I.M. Isaacs,Finite group theory, Amer. Math. Soc., Providence, R.I., 2008

work page 2008