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REVIEW 3 major objections 4 minor 35 references

Arithmetic invariants for finite simple and related groups

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Pith's one-line read Multiplicity-counted tuples of element orders, class sizes, and character degrees are claimed to form full invariants for finite simple groups and symmetric groups.

desk verdict Useful packaging of known results into one notation, but the central identity in Section 2 is false and Theorem 2.5 is unproven as written; a one-line fix should repair it. read the letter →

arxiv 2608.12783 v1 pith:RKFMDHJZ submitted 2026-08-13 math.GR

Andrey V. Vasil'ev This is my paper · ORCID
classification math.GR MSC 20D0620D6020C1520E45
keywords finitesimplegroupselementordersconjugacyclasssizescharacterdegreesfullsystemofinvariantssymmetricquasisimpleShiconjecture
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

This note proposes a uniform arithmetic language for the three classical datasets attached to a finite group: element orders, conjugacy class sizes, and irreducible character degrees, each considered with or without multiplicities and indexed by the divisors of the group order. In that language the paper claims that the multiplicity-counted tuple alone is a full system of invariants for finite simple groups, and also for symmetric groups; it observes that Shi's conjecture is exactly the statement that the order together with the element-order indicator tuple is full for simple groups. The framework is meant to make the Shi, Thompson, and Huppert recognition conjectures comparable instances of one question, and to turn open problems about related groups into concrete questions about which tuples separate isomorphism classes.

What carries the argument

The central object is the delta-invariant tuple. For $\mathrm{inv} \in \{\mathrm{eo}, \mathrm{cs}, \mathrm{cd}\}$, list the divisors $d_i$ of $n(G)$ in increasing order and put $\delta^{\mathrm{inv}}_i(G) = 1$ if $d_i$ occurs in the corresponding set and $0$ otherwise; for $\mathrm{inv}^*$ replace this by the multiplicity of $d_i$ in the multiset. The paper's intended bridge is the assertion that $n(G)$ is recovered from the multiplicity tuples by $n(G) = \sum_i \delta^{\mathrm{eo}^*}_i(G) = \sum_i \delta^{\mathrm{cs}^*}_i(G) = \sum_i (\delta^{\mathrm{cd}^*}_i(G))^2$, so that $\delta^{\mathrm{inv}^*}(G)$ determines $(n(G), \delta^{\mathrm{inv}}(G))$; the fullness results then reduce to known recognition theorems expressed in this vector language.

What would settle it

Direct computation settles the claimed bridge: for the symmetric group $S_3$, $\delta^{\mathrm{cs}^*}$ has one class of each size $1$, $2$, and $3$, so $\sum_i \delta^{\mathrm{cs}^*}_i = 3$ while $|S_3| = 6$; for the cyclic group $C_2$, $\delta^{\mathrm{cd}^*}$ has two entries equal to $1$, so $\sum_i (\delta^{\mathrm{cd}^*}_i)^2 = 4$ while $|C_2| = 2$. Since $C_2$ is a simple group, the displayed identity in Section 2 cannot be used to derive Theorem 2.5, and the fullness assertion would need a different proof to stand.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that for $\mathrm{inv}^* \in \{\mathrm{eo}^*, \mathrm{cs}^*, \mathrm{cd}^*\}$, the divisor-indexed multiplicity tuple $\delta^{\mathrm{inv}^*}(G)$ is a full system of invariants within the class of finite simple groups (Theorem 2.5) and within the symmetric groups (Theorem 3.4). The same formalism recasts Shi's conjecture as fullness of the pair $(n(G), \delta^{\mathrm{eo}}(G))$, and the paper reports that $(n(G), \delta^{\mathrm{cs}}(G))$ is full for simple and for alternating and symmetric groups, while $(n(G), \delta^{\mathrm{cd}}(G))$ remains open for simple groups. The note also surveys what is known for almost simple, quasisimple, and almost quasisimple groups, including fullness of $\delta^{\mathrm{cd}^*}$ for quasisimple groups and counterexamples showing that element-order data alone cannot separate some groups related to $A_6$.

Load-bearing premise

The load-bearing premise, displayed in Section 2, is that the multiplicity tuples determine the order of $G$ through the identities $n(G) = \sum_i \delta^{\mathrm{cs}^*}_i(G) = \sum_i (\delta^{\mathrm{cd}^*}_i(G))^2$; as written those identities are false, because the first sum counts conjugacy classes rather than elements and the second sums the squares of class counts of character degrees rather than the degrees themselves.

Editorial extensions

If this is right

  • For every finite simple group $L$, any finite group $G$ with the same element-order multiplicity tuple as $L$ would have to be isomorphic to $L$, and the analogous statement would hold for conjugacy-class-size and character-degree multiplicity tuples.
  • For every symmetric group $S_m$, the same uniqueness would hold under each of the three multiplicity tuples, extending the known characterization by character degrees alone.
  • Shi's conjecture would be exactly the statement that order plus the element-order indicator tuple separates simple groups, giving a common formulation under which Thompson's and Huppert's conjectures can be compared.
  • For quasisimple groups, fullness of the character-degree multiplicity tuple would mean that the complex group algebra determines the group, answering a question that originates in representation theory.
  • The open status of $(n, \delta^{\mathrm{cd}})$ for simple groups would be located as the one missing piece in the uniform framework, with Huppert's conjecture sufficient but not necessary for a positive answer.

Reading between the lines

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

  • If the intended bridge can be repaired by replacing the sums with $n = \sum_i d_i \delta^{\mathrm{cs}^*}_i(G)$ and $n = \sum_i d_i^2 \delta^{\mathrm{cd}^*}_i(G)$, the same framework would survive; checking which of Theorems 2.3, 2.5, and 3.4 remain derivable is a direct next step.
  • One could test the separating power of these tuples computationally over small groups: for each order up to some bound, ask whether any two non-isomorphic groups share the same $\delta^{\mathrm{inv}^*}$ tuple, which would give a low-cost empirical check of the spirit of Theorem 2.5 before a repaired proof appears.
  • The paper's distinction between the minimal input $(n, \delta^{\mathrm{inv}})$ and the maximal input $\delta^{\mathrm{inv}^*}$ suggests a natural interpolation problem: for which groups does a partial multiplicity tuple, say only the first few divisors, already separate isomorphism classes?
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces a uniform formalism for arithmetic invariants of finite groups: for each invariant family inv in {eo, cs, cd}, the tuple δinv(G) records, for each divisor of |G|, whether that divisor occurs in the corresponding set, while the starred tuple δinv*(G) records the multiplicity of each divisor in the corresponding multiset. The paper's central assertions are that, for simple groups and for symmetric groups, the starred tuples δinv*(G) form full systems of invariants (Theorems 2.5 and 3.4), obtained by reducing starred data to the order n(G) together with the unstarred tuple. It also restates Shi's conjecture in this language, reviews the status of Thompson's and Huppert's conjectures, discusses related groups, and poses several open problems, including a conjecture about almost quasisimple groups.

Significance. If the main claims are correct, the paper provides a genuinely useful unifying perspective: a single framework that packages element-order, conjugacy-class-size, and character-degree invariants with and without multiplicities, and that makes precise the sense in which multiplicity data is stronger than set data. The identification of Shi's conjecture as the fullness of (n(G), δeo(G)) and the intended transfer principle from δinv*(G) to (n(G), δinv(G)) are valuable organizational ideas. The survey of known results and open problems is also useful. The paper's strengths are the clarity of the proposed notation and the fact that the main reduction, once corrected as described below, is elementary and directly checkable. The present proof of Theorem 2.5, however, contains an incorrect identity, so the central claim is currently not established as written.

major comments (3)
  1. [Section 2, displayed identity after 'Since'] The displayed identity n(G) = Σ_i δcs*_i(G) = Σ_i (δcd*_i(G))^2 is false. By the definitions in the same section, Σ_i δcs*_i(G) is the number of conjugacy classes of G, not the order |G|, and Σ_i (δcd*_i(G))^2 is a sum of squared multiplicities of character degrees, not generally equal to |G|. The correct identities are n(G) = Σ_i d_i δcs*_i(G) and n(G) = Σ_i d_i^2 δcd*_i(G), where d_i runs over the divisors of n(G). Because the tuples are indexed by the labelled divisors d_i, the conclusion that δinv*(G) determines (n(G), δinv(G)) is still recoverable for inv* = cs* and cd* if these weighted formulas are used. As printed, however, the proof of Theorem 2.5 is invalid and must be rewritten; the identity for eo* is correct.
  2. [Theorems 2.2, 2.3 and the cs* branch of Theorem 2.5] The cs* results depend on the author's own preprint [14], including [14, Lemma 2.4], but the needed statement and proof are not reproduced in this note. Consequently, Theorem 2.3 and the cs* assertion of Theorem 2.5 are conditional on [14] being correct and publicly available. The paper should either state and prove the necessary lemma from [14] or explicitly flag these results as conditional on that preprint.
  3. [Theorem 2.5, cd* branch] The sentence 'it follows from [28,30,31] that δcd*(G) is [full]' is not accompanied by a precise statement of what those papers prove. Since Huppert's conjecture is open for generic classical groups, and since the passage from 'determined by character degrees' to 'fullness of δcd*' is not automatic without a statement about multiplicities, the reader cannot verify the cd* branch of Theorem 2.5 from the text. Please state the exact cited theorems and explain how they imply fullness of the starred tuple in the class of all finite groups.
minor comments (4)
  1. [Definition of δinv*(G)] The tuple δinv*(G) is defined as a vector of length τ(n), so its length depends on the group. To make it a well-defined invariant on the class of all finite groups, the paper should either define it as a function on all positive divisors (with value 0 outside D(n)) or explicitly fix a common index set; otherwise equality of tuples of different lengths is not formally defined.
  2. [Problem 3.9] In Problem 3.9(i), 'sovbale' should be 'solvable'.
  3. [References [10], [11], and [14]] The phrase 'Thompsons's conjecture' should be 'Thompson's conjecture', and the title of [14] misspells 'conjugacy' as 'cojugacy'.
  4. [Remark after Conjecture 3.8] The example with the two maximal subgroups of M23 is asserted via [33, Section 4.3] but is not made self-contained; stating the common multiset of element orders would strengthen the illustrative point.

Circularity Check

1 steps flagged · score 4.0 of 10

One load-bearing self-citation ([14]) carries the cs* branch; the eo* and cd* branches rest on independent external results, and the flawed sum identity is a repairable proof gap, not circularity.

  1. self citation load bearing [Section 2, Theorems 2.2–2.3, and reference [14]]
    "Recently in [14], we proved Theorem 2.2. The system of invariants (n(G),δ cs(G)) is full for the alternating and symmetric groups. From the validity of Thompson’s conjecture for all simple groups except the alternating groups and [14, Lemma 2.4], it follows that the system of invariants is full for all simple groups (see also [14, Corollary 1.3])."

    Theorem 2.3, and with it the cs* case of Theorem 2.5, is not proved in this note but is imported from the authors' own unpublished arXiv preprint [14]. The alternating and symmetric cs* characterization is exactly the gap that Thompson's conjecture leaves open (for alternating groups it is only known modulo the binary Goldbach conjecture, [10]), so [14] is load-bearing rather than incidental. Because [14] is not reproduced, machine-checked, or independently verified in the manuscript, the derivation chain for the cs* branch reduces to the authors' own unverified prior work. The eo* and cd* branches of Theorem 2.5 are not affected: they use Shi's theorem and Tong-Viet's character-degree results.

full rationale

The paper is best read as a survey/unified notation: Shi's conjecture is restated as Theorem 2.1, and Theorems 2.5 and 3.4 assemble known results into the delta-invariant language. For eo*, the identity n(G)=sum_i δ^{eo*}_i is correct by definition, and fullness follows from Shi's conjecture [34] and Tong-Viet's cd* results. For cd*, δ^{cd*} fullness for simple and symmetric groups is cited to external work [28,29,30,31]. The only load-bearing self-citation is [14], by Gorshkov and Vasil'ev, which supplies the cs* fullness for alternating and symmetric groups; Theorem 2.3 and the cs* branch of Theorem 2.5 inherit this unpublished result. That is a genuine self-citation concern but not a by-construction equivalence. Separately, the displayed identity n(G)=sum δ^{cs*}_i = sum (δ^{cd*}_i)^2 is arithmetically false: the cs* sum is the number of conjugacy classes, and the cd* sum is not generally |G|. The intended conclusion that the tuple determines n is still recoverable from the class equation n=sum d_i δ^{cs*}_i and orthogonality n=sum d_i^2 δ^{cd*}_i, so this is a correctable proof gap rather than a case of the result being equivalent to its inputs. No other circular reductions appear. Overall score 4 reflects one load-bearing self-citation in an otherwise externally grounded framework.

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

The paper introduces no free parameters or new entities. It relies on established results and on the classification of finite simple groups. The main unproved assumption is the conditional status of Thompson's conjecture for alternating groups, which affects Theorem 2.3.

assumptions (2)
  • domain assumption Classification of finite simple groups (CFSG)
    The paper relies on the complete list of finite simple groups (Section 1) as background for all statements about simple groups.
  • domain assumption Validity of Thompson's conjecture for the alternating groups (or a substitute from [14])
    Theorem 2.3 states that (n(G), delta^{cs}(G)) is full for simple groups. This relies on Thompson's conjecture for all simple groups; for the alternating groups the conjecture is known only conditionally (modulo binary Goldbach) in [10], and the paper cites [14, Lemma 2.4] as a replacement without providing the proof.

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Pith. "Pith review of Arithmetic invariants for finite simple and related groups." pith.science (2026). https://pith.science/paper/RKFMDHJZ

@misc{pith2026260812783,
  author       = {Pith},
  title        = {Pith review of: Arithmetic invariants for finite simple and related groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKFMDHJZ}},
  note         = {Machine review of arXiv:2608.12783}
}
read the original abstract

In this short note we address the problems of characterization of simple and related groups by various arithmetic invariants. We come with some uniform way to think about such sort of questions and discuss what has been already done and what we still do not know but wish to know in this field.

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Works this paper leans on

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