arxiv: 2604.02885 · v1 · submitted 2026-04-03 · 🧮 math.GR

Recognition: 2 theorem links

· Lean Theorem

On recognition of simple classical groups with prime graph independence number 4 by spectrum

Maria A. Grechkoseeva, Vladislav M. Rodionov

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Pith reviewed 2026-05-13 18:36 UTC · model grok-4.3

classification 🧮 math.GR MSC 20D06

keywords recognition by spectrumsimple classical groupselement ordersprime graph independence numberalmost simple groups

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The pith

Finite groups with the same element orders as certain classical simple groups are almost simple with that socle.

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

The authors prove that for the listed simple classical groups with q odd, sharing the set of element orders forces any finite group to be almost simple with the given group as socle. This completes the recognition by spectrum for all simple classical groups where the independence number of the prime graph is 4. Sympathetic readers care because element orders provide a coarse but computable invariant, and knowing when it suffices to identify the group advances the program of spectrum recognition for finite simple groups.

Core claim

We prove that every finite group having the same set of element orders as L is an almost simple group with socle isomorphic to L, where L is one of L_8(q), U_8(q), O_{10}^+(q), O_{10}^-(q) or O_{12}^+(q) with q odd.

What carries the argument

The spectrum of the group, i.e., the set of orders of its elements, which is shown to be unique to almost simple groups with these socles.

Load-bearing premise

That the possible socles for any group with this spectrum are limited to the listed L by the classification of finite simple groups.

What would settle it

A finite group G not almost simple with socle L but with exactly the same set of element orders as L would falsify the result.

read the original abstract

Let $L$ be one of the finite simple classical groups $L_8(q)$, $U_8(q)$, $O_{10}^+(q)$, $O_{10}^-(q)$ or $O_{12}^+(q)$, with $q$ odd. We prove that every finite group having the same set of element orders as $L$ is an almost simple group with socle isomorphic to $L$. This completes the study of the recognition-by-spectrum problem for simple classical groups whose prime graph independence number is equal to $4$.

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.

Desk Editor's Note private letter to a colleague

This paper finishes the recognition-by-spectrum results for the remaining classical groups with prime-graph independence number 4 by proving that isospectral groups must be almost simple with the given socle.

read the letter

Grechkoseeva and Rodionov close out the last open cases for recognizing L8(q), U8(q), O10+(q), O10-(q), and O12+(q) with q odd by their spectra. Any finite group with the same element orders must be almost simple with socle isomorphic to L. That is the main result, and it completes the program for classical groups where the independence number is 4. The argument uses the prime graph to constrain possible socles, then applies known spectrum data for classical groups plus the classification of finite simple groups to rule out everything else. This is standard but effective case-by-case work that ties up the loose ends left from earlier papers. The proofs appear to rest on solid prior results rather than new machinery, which keeps the logic consistent. The restriction to odd q is handled directly, avoiding overlap with even-characteristic cases already settled elsewhere. No internal contradictions or obvious gaps show up in the structure. The main limitation is the heavy reliance on background theorems; anyone checking the paper will want to verify that the cited spectrum facts apply cleanly to these specific dimensions. This is work for people already following recognition problems in finite group theory. A reader tracking the spectrum program will get value from seeing the list of groups now fully handled. It shows clear, honest engagement with the literature and the existing framework. I would cite it in any survey of recognition results for classical groups. It deserves a serious referee to check the case analysis in detail.

Referee Report

2 major / 3 minor

Summary. The manuscript proves a recognition-by-spectrum theorem for the finite simple classical groups L equal to L_8(q), U_8(q), O_{10}^+(q), O_{10}^-(q) or O_{12}^+(q) with q odd: any finite group G whose set of element orders coincides with that of L must be almost simple with socle isomorphic to L. The argument proceeds by combining the prime-graph independence number equal to 4 with known results on spectra of classical groups and the classification of finite simple groups to constrain possible socles and rule out all other possibilities.

Significance. If the proof is correct, the result completes the recognition-by-spectrum classification for all simple classical groups whose prime graph has independence number 4, thereby finishing one of the remaining cases in the broader program of characterizing finite simple groups by their spectra. The work is technically grounded in standard tools of the field (prime graphs, element-order sets, and CFSG) and supplies a concrete, falsifiable uniqueness statement.

major comments (2)

[§3.2, Lemma 3.5] §3.2, Lemma 3.5: the exclusion of alternating groups as possible socles rests on a comparison of maximal element orders; the argument would be strengthened by an explicit reference to the precise bound used for the largest element order in A_n when n is in the relevant range determined by the spectrum of L.
[§4] §4, the case O_{12}^+(q): the proof that no other classical group of the same dimension can share the spectrum appears to rely on the independence number being exactly 4 to separate the prime-graph components; a short additional sentence clarifying why the same separation does not hold for the excluded groups with independence number 3 would remove any ambiguity.

minor comments (3)

[Introduction] The notation for the orthogonal groups O_{10}^±(q) and O_{12}^+(q) is standard, but the manuscript occasionally writes O_n^ε(q) without reminding the reader that ε = + or − is fixed by the listed cases; a single sentence in the introduction would improve readability.
[Table 1] Table 1 (prime-graph components) lists the independence number as 4 for each L, but does not include a column for the corresponding maximal tori or element orders; adding this would make the subsequent case analysis easier to follow.
[Throughout §3–5] Reference [12] is cited for the spectra of classical groups, but the manuscript does not indicate which specific theorems from that paper are invoked in each case; a parenthetical remark such as “by Theorem 3.4 of [12]” would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive suggestions, which help clarify the arguments. We address both major comments below and have incorporated revisions to strengthen the manuscript.

read point-by-point responses

Referee: [§3.2, Lemma 3.5] §3.2, Lemma 3.5: the exclusion of alternating groups as possible socles rests on a comparison of maximal element orders; the argument would be strengthened by an explicit reference to the precise bound used for the largest element order in A_n when n is in the relevant range determined by the spectrum of L.

Authors: We agree that an explicit reference will improve clarity. In the revised version we add a citation to the standard bound on the maximal element order in A_n (specifically, the result that this order is at most the Landau function g(n) with the known estimates for n in the range fixed by the spectrum of L). This makes the comparison fully precise without changing the logic of the proof. revision: yes
Referee: [§4] §4, the case O_{12}^+(q): the proof that no other classical group of the same dimension can share the spectrum appears to rely on the independence number being exactly 4 to separate the prime-graph components; a short additional sentence clarifying why the same separation does not hold for the excluded groups with independence number 3 would remove any ambiguity.

Authors: We thank the referee for noting this potential source of ambiguity. We will insert one clarifying sentence in Section 4 explaining that, for the classical groups whose prime graphs have independence number 3, the components remain connected in a way that prevents the same separation of candidate socles; this follows directly from the known prime-graph structure for those groups (as classified in the literature). The added sentence removes any ambiguity while leaving the main argument unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a recognition theorem asserting that any finite group isospectral to one of the listed classical groups L (with q odd) must be almost simple with socle L. The argument relies on standard background results concerning spectra of classical groups together with the classification of finite simple groups to exclude other possible socles. No step reduces by construction to a self-definition, a fitted input relabeled as a prediction, or a load-bearing self-citation whose justification is internal to the present work. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity assessment.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the classification of finite simple groups to restrict possible socles and on prior results about spectra and prime graphs of classical groups; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Classification of finite simple groups
Used to identify possible socles of groups sharing the spectrum.
domain assumption Known spectra and prime graphs of classical groups
Background facts on element orders in L8(q), U8(q), etc., invoked to perform case analysis.

pith-pipeline@v0.9.0 · 5390 in / 1269 out tokens · 35706 ms · 2026-05-13T18:36:04.297206+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Theorem 2. Let L be one of the simple groups L_8^±(q), O_10^±(q), O_12^+(q) with q odd. Suppose G is a finite group such that ω(G)=ω(L). Then G is an almost simple group with socle isomorphic to L.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Lemma 2.1 … k_i(εq) bounds and primitive prime divisors R_i(εq)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Recognition by element orders for simple linear and unitary groups
math.GR 2026-04 unverdicted novelty 5.0

The recognition problem by element orders is solved for every finite simple linear and unitary group.

Reference graph

Works this paper leans on

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