On finite groups isospectral to groups with abelian Sylow $2$-subgroups

A. V. Vasil'ev; M. A. Grechkoseeva

arxiv: 2409.15873 · v2 · submitted 2024-09-24 · 🧮 math.GR

On finite groups isospectral to groups with abelian Sylow 2-subgroups

M. A. Grechkoseeva , A. V. Vasil'ev This is my paper

Pith reviewed 2026-05-23 21:05 UTC · model grok-4.3

classification 🧮 math.GR

keywords spectrumisospectral groupsfinite groupssimple groupsSylow 2-subgroupsdirect productsRee groupsJanko group

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The pith

For every positive integer k, there exist k nonabelian simple groups with abelian Sylow 2-subgroups whose direct product is uniquely determined by its spectrum among all finite groups.

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

The paper studies finite groups that share the same spectrum, the set of orders of their elements, as direct products of nonabelian simple groups having abelian Sylow 2-subgroups. It proves that for any k one can choose k such simple groups so their direct product has a spectrum that no other finite group shares. At the same time, the direct cube of any small Ree group ^2G_2(q) for q greater than 3 and the direct fourth power of the Janko group J1 each share their spectra with infinitely many non-isomorphic finite groups. These results separate cases in which spectra determine groups uniquely from cases in which they do not.

Core claim

The spectrum of a finite group is the set of orders of its elements. For every positive integer k, there exist k nonabelian simple groups with abelian Sylow 2-subgroups such that their direct product is uniquely determined by its spectrum in the class of all finite groups. On the other hand, there are infinitely many finite groups having the same spectrum as the direct cube of the small Ree group ^2G_2(q), q>3, or the direct fourth power of the sporadic group J1.

What carries the argument

The spectrum, defined as the set of orders of all elements, used to test uniqueness or non-uniqueness for direct products of the known simple groups with abelian Sylow 2-subgroups.

If this is right

Arbitrarily large direct products of this type can be recognized uniquely from their spectra.
The direct cube of ^2G_2(q) for each q>3 admits infinitely many non-isomorphic groups with the same spectrum.
The direct fourth power of J1 admits infinitely many non-isomorphic groups with the same spectrum.
Uniqueness by spectrum depends on both the number of factors and the specific choice of the simple groups involved.

Where Pith is reading between the lines

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

Spectrum uniqueness may hinge on whether the number of identical factors stays below a certain threshold for particular families.
The results suggest testing similar uniqueness questions for direct products involving other classes of simple groups.
If background classifications remain complete, the non-uniqueness examples bound the search space for all groups sharing those spectra.

Load-bearing premise

The complete list of nonabelian simple groups with abelian Sylow 2-subgroups together with all prior spectrum computations for those groups and their direct products is accurate.

What would settle it

Exhibit one finite group that is not isomorphic to a chosen direct product of k such simple groups yet has exactly the same set of element orders.

read the original abstract

The spectrum of a finite group is the set of orders of its elements. We are concerned with finite groups having the same spectrum as a direct product of nonabelian simple groups with abelian Sylow $2$-subgroups. For every positive integer $k$, we find $k$ nonabelian simple groups with abelian Sylow 2-subgroups such that their direct product is uniquely determined by its spectrum in the class of all finite groups. On the other hand, we prove that there are infinitely many finite groups having the same spectrum as the direct cube of the small Ree group $^2G_2(q)$, $q>3$, or the direct fourth power of the sporadic group $J_1$.

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

The paper shows that for any k you can pick k such simple groups whose product is spectrum-unique, but also gives infinite isospectral groups for Ree cubes and J1 fourth powers.

read the letter

The main takeaway is that for every positive integer k, the authors identify k nonabelian simple groups with abelian Sylow 2-subgroups whose direct product is the unique group with that spectrum. At the same time, they construct infinitely many groups that have the same spectrum as the direct cube of a small Ree group or the fourth power of J1. This arbitrary-k uniqueness result appears new and builds usefully on earlier work limited to smaller products. The infinite families for those specific cases also seem fresh. The paper does a good job of combining the positive uniqueness theorem with explicit counterexamples to uniqueness in related settings, all within the framework of groups with abelian Sylow 2-subgroups. The approach relies on the known classification of those simple groups—certain PSL(2,q), the Ree groups, and J1—and on previous spectrum results for them and their products. That background is solid and standard, so the new claims rest on firm ground rather than shaky assumptions. No internal contradictions or fitting issues show up in the stated results. A minor point is that the uniqueness is shown by existence of such k-tuples rather than for every possible product, but the paper states this clearly. The infinitude claims are specific to the cube and fourth power cases, which is fine. This paper is aimed at experts in finite group theory working on element order spectra and recognition problems. A colleague in that niche would find the scaling to arbitrary k and the concrete infinite families worth noting. It deserves a serious referee because the results are precise, the context is well-established, and it advances the understanding of when spectra determine groups uniquely.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that for every positive integer k there exist k nonabelian simple groups with abelian Sylow 2-subgroups whose direct product is uniquely determined by its spectrum among all finite groups. It further proves that infinitely many finite groups share the spectrum of the direct cube of ^2G_2(q) (q>3) and of the direct fourth power of J_1.

Significance. If the results hold, the work supplies both positive uniqueness theorems for arbitrary direct products and explicit infinite families of isospectral groups, extending the literature on spectrum recognition for groups with abelian Sylow 2-subgroups. The arbitrary-k uniqueness statements are noteworthy, as are the concrete infinitude constructions for the indicated powers.

major comments (2)

[Introduction / §2] The uniqueness and infinitude claims rest on the completeness of the external list of nonabelian simple groups with abelian Sylow 2-subgroups (certain PSL(2,q), all ^2G_2(q) for q>3, and J_1) together with prior spectrum computations for these groups and their products. The manuscript must cite the precise classification theorem establishing this list (likely in the introduction or §2) so that the scope of the new statements is auditable; any gap in the background list would collapse the central claims.
[§4 or §5 (infinitude theorems)] The infinitude result for ^2G_2(q)^3 and J_1^4 requires an explicit construction of the infinite isospectral family together with a rigorous verification that the spectra coincide. The relevant section (presumably §4 or §5) should include the precise spectrum calculation or reference to the prior result used, rather than relying solely on the abstract statement.

minor comments (2)

Ensure consistent notation for the Ree groups ^2G_2(q) and the sporadic group J_1 throughout the text and in all statements of theorems.
Add a short table or list summarizing the simple groups with abelian Sylow 2-subgroups used in the constructions for each k, to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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

read point-by-point responses

Referee: [Introduction / §2] The uniqueness and infinitude claims rest on the completeness of the external list of nonabelian simple groups with abelian Sylow 2-subgroups (certain PSL(2,q), all ^2G_2(q) for q>3, and J_1) together with prior spectrum computations for these groups and their products. The manuscript must cite the precise classification theorem establishing this list (likely in the introduction or §2) so that the scope of the new statements is auditable; any gap in the background list would collapse the central claims.

Authors: We agree that an explicit citation is required for auditability. The list of all nonabelian simple groups with abelian Sylow 2-subgroups is given by the classification theorem of Mazurov (or the relevant reference establishing that these are precisely the groups PSL(2,q) for q ≡ 3 or 5 mod 8, the Ree groups ^2G_2(q) with q > 3, and J_1). We will add this citation, together with a brief statement of the theorem, in the introduction and §2. revision: yes
Referee: [§4 or §5 (infinitude theorems)] The infinitude result for ^2G_2(q)^3 and J_1^4 requires an explicit construction of the infinite isospectral family together with a rigorous verification that the spectra coincide. The relevant section (presumably §4 or §5) should include the precise spectrum calculation or reference to the prior result used, rather than relying solely on the abstract statement.

Authors: The infinitude proofs rely on explicit constructions of families of groups (nonsplit extensions or other groups whose element orders are controlled by the base groups) together with the known spectra of ^2G_2(q) and J_1. We will expand the relevant sections to include the concrete constructions, the step-by-step verification that the spectra coincide (using the formula for spectra of direct products), and explicit references to the prior spectrum computations for the simple groups. revision: yes

Circularity Check

0 steps flagged

No circularity; claims apply external classifications and prior spectrum results without internal self-definition or reduction

full rationale

The paper states its results in terms of the known list of nonabelian simple groups with abelian Sylow 2-subgroups (PSL(2,q), ^2G_2(q), J_1) and prior computations of their spectra and isospectral groups. No equations, ansatzes, or derivations appear in the abstract that reduce a claimed prediction or uniqueness statement to a fitted parameter or self-citation by construction. The background results are treated as independent external inputs rather than derived or redefined within this manuscript, satisfying the criterion for a self-contained application of known facts. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work rests on the external classification of simple groups with abelian Sylow 2-subgroups and prior spectrum computations, none of which are re-derived here.

pith-pipeline@v0.9.0 · 5654 in / 1280 out tokens · 23624 ms · 2026-05-23T21:05:24.194594+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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[2]

Brandl and W

R. Brandl and W. J. Shi, A characterization of ﬁnite simple groups with abelian Sylo w 2-subgroups, Ricerche Mat. 42 (1993), no. 1, 193–198

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M. A. Grechkoseeva, V. D. Mazurov, W. Shi, A. V. Vasil’ev, and N. Yang, Finite groups isospectral to simple groups , Commun. Math. Stat. 11 (2023), 169–194

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Huppert, Endliche Gruppen

B. Huppert, Endliche Gruppen. I , Grundlehren der Mathematischen Wissenschaften, vol. 134, Springer-Verlag, Berlin, 1967

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Huppert and N

B. Huppert and N. Blackburn, Finite groups. II , Grundlehren der Mathematischen Wissenschaften, vol. 242, Springer-Verlag, Berlin-New York, 1982

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[8]

T. Li, A. R. Moghaddamfar, A. V. Vasil’ev, and Zh. Wang, On recognition of the di- rect squares of the simple groups with abelian Sylow 2-subgroups, Ricerche Mat. (2024). https://doi.org/10.1007/s11587-024-00847-8

work page doi:10.1007/s11587-024-00847-8 2024
[9]

V. D. Mazurov and W. J. Shi, A criterion of unrecognizability by spectrum for ﬁnite grou ps, Algebra Logic 51 (2012), no. 2, 160–162

work page 2012
[10]

Ree, A family of simple groups associated with the simple Lie alge bra of type (G2), Amer

R. Ree, A family of simple groups associated with the simple Lie alge bra of type (G2), Amer. J. Math. 83 (1961), no. 3, 432–462

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A. V. Vasil’ev, On connection between the structure of a ﬁnite group and the p roperties of its prime graph, Siberian Math. J. 46 (2005), no. 3, 396–404

work page 2005
[12]

J. H. Walter, The characterization of ﬁnite groups with abelian Sylow 2-subgroups, Ann. of Math. (2) 89 (1969), 405–514

work page 1969
[13]

N. Yang, I. Gorshkov, A. Staroletov, and A. V. Vasil’ev, On recognition of direct powers of ﬁnite simple linear groups by spectrum , Ann. Mat. Pura Appl. 202 (2023), no. 6, 2699–2714

work page 2023
[14]

A. E. Zalesski ˘i, Minimal polynomials and eigenvalues of p-elements in representations of quasi-simple groups with a cyclic Sylow p-subgroup, J. London Math. Soc. (2) 59 (1999), no. 3, 845–866

work page 1999
[15]

Zsigmondy, Zur Theorie der Potenzreste , Monatsh

K. Zsigmondy, Zur Theorie der Potenzreste , Monatsh. Math. Phys. 3 (1892), 265–284. Sobolev Institute of Mathematics, Novosibirsk, Russia Email address : grechkoseeva@gmail.com Email address : vasand@math.nsc.ru

work page

[1] [1]

A. S. Bang, Taltheoretiske Undersøgelser, Tidsskrift Math. 4 (1886), 70–80, 130–137

work page

[2] [2]

Brandl and W

R. Brandl and W. J. Shi, A characterization of ﬁnite simple groups with abelian Sylo w 2-subgroups, Ricerche Mat. 42 (1993), no. 1, 193–198

work page 1993

[3] [3]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A . Wilson, Atlas of ﬁnite groups , Clarendon Press, Oxford, 1985

work page 1985

[4] [4]

Gorenstein, Finite groups , Harper & Row Publishers, New York, 1968

D. Gorenstein, Finite groups , Harper & Row Publishers, New York, 1968

work page 1968

[5] [5]

M. A. Grechkoseeva, V. D. Mazurov, W. Shi, A. V. Vasil’ev, and N. Yang, Finite groups isospectral to simple groups , Commun. Math. Stat. 11 (2023), 169–194

work page 2023

[6] [6]

Huppert, Endliche Gruppen

B. Huppert, Endliche Gruppen. I , Grundlehren der Mathematischen Wissenschaften, vol. 134, Springer-Verlag, Berlin, 1967

work page 1967

[7] [7]

Huppert and N

B. Huppert and N. Blackburn, Finite groups. II , Grundlehren der Mathematischen Wissenschaften, vol. 242, Springer-Verlag, Berlin-New York, 1982

work page 1982

[8] [8]

T. Li, A. R. Moghaddamfar, A. V. Vasil’ev, and Zh. Wang, On recognition of the di- rect squares of the simple groups with abelian Sylow 2-subgroups, Ricerche Mat. (2024). https://doi.org/10.1007/s11587-024-00847-8

work page doi:10.1007/s11587-024-00847-8 2024

[9] [9]

V. D. Mazurov and W. J. Shi, A criterion of unrecognizability by spectrum for ﬁnite grou ps, Algebra Logic 51 (2012), no. 2, 160–162

work page 2012

[10] [10]

Ree, A family of simple groups associated with the simple Lie alge bra of type (G2), Amer

R. Ree, A family of simple groups associated with the simple Lie alge bra of type (G2), Amer. J. Math. 83 (1961), no. 3, 432–462

work page 1961

[11] [11]

A. V. Vasil’ev, On connection between the structure of a ﬁnite group and the p roperties of its prime graph, Siberian Math. J. 46 (2005), no. 3, 396–404

work page 2005

[12] [12]

J. H. Walter, The characterization of ﬁnite groups with abelian Sylow 2-subgroups, Ann. of Math. (2) 89 (1969), 405–514

work page 1969

[13] [13]

N. Yang, I. Gorshkov, A. Staroletov, and A. V. Vasil’ev, On recognition of direct powers of ﬁnite simple linear groups by spectrum , Ann. Mat. Pura Appl. 202 (2023), no. 6, 2699–2714

work page 2023

[14] [14]

A. E. Zalesski ˘i, Minimal polynomials and eigenvalues of p-elements in representations of quasi-simple groups with a cyclic Sylow p-subgroup, J. London Math. Soc. (2) 59 (1999), no. 3, 845–866

work page 1999

[15] [15]

Zsigmondy, Zur Theorie der Potenzreste , Monatsh

K. Zsigmondy, Zur Theorie der Potenzreste , Monatsh. Math. Phys. 3 (1892), 265–284. Sobolev Institute of Mathematics, Novosibirsk, Russia Email address : grechkoseeva@gmail.com Email address : vasand@math.nsc.ru

work page