On Independence Number of Comaximal Subgroup Graph

Angsuman Das; Arnab Mandal

REVIEW 4 major objections 4 minor 15 references

On Independence Number of Comaximal Subgroup Graph

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that any finite group with comaximal subgroup graph independence number at most 51 is solvable, that A5 is uniquely recovered from its graph, and that smaller bounds force supersolvability and nilpotency up to exceptions.

desk verdict Sharp thresholds, plausible results, but the main solvability proof rests on an unverified SAGE assertion over infinite families. read the letter →

arxiv 2506.03848 v1 pith:VEVETOCN submitted 2025-06-04 math.GR

Angsuman Das , Arnab Mandal This is my paper

classification math.GR MSC 05C2520D1020D15

keywords comaximalsubgroupgraphindependencenumbersolvablegroupssupersolvablenilpotentminimalsimplealternatinggroupA5sharpthresholds

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 proves that a single graph statistic, the independence number of the comaximal subgroup graph, sharply separates solvable from nonsolvable finite groups. The graph's vertices are the nontrivial proper subgroups of a group $G$, with two vertices joined when their product is all of $G$, and the independence number counts the largest family of such subgroups with no two adjacent. The central result is that $\alpha(\Gamma(G))\le 51$ forces $G$ to be solvable, and the bound is sharp because the smallest nonsolvable group $A_5$ has $\alpha=52$ and is itself uniquely recoverable from its graph. At lower thresholds the same statistic forces supersolvability ($\alpha\le 14$, with three exceptions) and nilpotency ($\alpha\le 6$, with five exceptions), and both bounds are shown to be sharp.

What carries the argument

The central object is the comaximal subgroup graph $\Gamma(G)$, whose vertices are the nontrivial proper subgroups of $G$ and whose edges join $H$ and $K$ when $HK=G$; the independence number $\alpha$ counts the largest set of vertices with no edges among them. The main mechanism is a minimal-counterexample reduction: a nonsolvable $G$ with $\alpha\le 51$ would have every proper subgroup solvable, hence $G$ would be a minimal simple group, and the classification of minimal simple groups restricts $G$ to five families, namely $\operatorname{PSL}(2,2^p)$, $\operatorname{PSL}(2,3^p)$, $\operatorname{PSL}(2,p)$ for primes $p$ with $5\mid p^2+1$, the groups $Sz(2^p)$, and $\operatorname{PSL}(3,3)$. The proof combines subgroup-count bounds (a classical theorem on maximal subgroups of solvable groups, a lower bound on the number of subgroups of non-cyclic $p$-groups, and a solvability criterion in terms of the number of subgroups) with a computer-algebra check to conclude $\alpha(\Gamma(G))\ge 52$ in all these cases; for the supersolvability and nilpotency thresholds, the same subgroup-counting and exhaustive enumeration of small orders produces the stated exceptional groups.

What would settle it

Compute $\alpha(\Gamma(G))$ for one member of each infinite family in the classification of minimal simple groups, for example $\operatorname{PSL}(2,2^p)$ with $p\ge 3$ or $Sz(2^p)$ with odd $p$. Any value $\le 51$ would disprove Theorem 2.1, while a complete reproducible table for the first several members of the families would either support the asserted lower bound or expose the missing case.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a sharp structural transition: the independence number of the comaximal subgroup graph is a group-theoretic classifier. Theorem 2.1 states that every finite group with $\alpha(\Gamma(G))\le 51$ is solvable, and Remark 2.1 records the sharpness example $\alpha(\Gamma(A_5))=52$; Theorem 2.2 states that $\Gamma(G)\cong\Gamma(A_5)$ forces $G\cong A_5$. Theorem 3.1 states that $\alpha\le 14$ forces supersolvability except for $A_4$, $\operatorname{SL}(2,3)$, and $(\mathbb{Z}_2\times\mathbb{Z}_2)\rtimes\mathbb{Z}_9$. Theorem 4.1 states that $\alpha\le 6$ forces nilpotency except for $S_3$, $D_5$, $\mathbb{Z}_3\rtimes\mathbb{Z}_4$, $\mathbb{Z}_3\rtimes\mathbb{Z}_8$, and $\mathbb{Z}_5\rtimes\mathbb{Z}_4$. The thresholds 14 and 6 are flanked by infinite families with $\alpha=15$ and $\alpha=7$, respectively, which is why the exceptions are finite but the bounds cannot be improved.

Load-bearing premise

The whole proof of Theorem 2.1 rests on the assertion that every group in the infinite list of minimal simple groups has $\alpha(\Gamma(G))\ge 52$, a claim supported only by a computer-algebra check that is neither written out nor reproduced; if even one listed group had $\alpha\le 51$, the theorem would fail.

Editorial extensions

If this is right

Every finite nonsolvable group has $\alpha(\Gamma(G))$ at least 52, so any finite group with $\alpha\le 51$ is certified solvable by its comaximal subgroup graph alone.
The graph $\Gamma(A_5)$, which is a complete bipartite graph with parts of sizes 5 and 12 plus 40 isolated vertices, determines $A_5$ up to isomorphism; this is false for $A_4$, whose graph is shared by non-isomorphic groups.
If $\alpha(\Gamma(G))\le 14$, then $G$ is supersolvable unless $G$ is one of $A_4$, $\operatorname{SL}(2,3)$, or $(\mathbb{Z}_2\times\mathbb{Z}_2)\rtimes\mathbb{Z}_9$.
If $\alpha(\Gamma(G))\le 6$, then $G$ is nilpotent unless $G$ is one of $S_3$, $D_5$, $\mathbb{Z}_3\rtimes\mathbb{Z}_4$, $\mathbb{Z}_3\rtimes\mathbb{Z}_8$, or $\mathbb{Z}_5\rtimes\mathbb{Z}_4$; in particular, $\alpha\le 6$ already forces supersolvability with no exceptions.
The thresholds are sharp: the infinite families $A_4\times\mathbb{Z}_p$ and $S_3\times\mathbb{Z}_p$ have independence numbers 15 and 7, respectively, so the 14 and 6 bounds cannot be raised.

Reading between the lines

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

Editorial inference: the same minimal-counterexample machinery could be used to test whether other graph parameters of $\Gamma(G)$, such as the domination number raised in the paper's open problems, exhibit the same phase-transition behavior at a small threshold.
Editorial inference: because the infinite-family lower bound $\alpha\ge 52$ is only asserted computationally, a natural strengthening is to derive an analytic lower bound on the number of maximal subgroups of $\operatorname{PSL}(2,2^p)$ and $Sz(2^p)$; that would also identify the extremal configurations responsible for $\alpha=52$.
Editorial inference: the two infinite sharpness families, $A_4\times\mathbb{Z}_p$ at $\alpha=15$ and $S_3\times\mathbb{Z}_p$ at $\alpha=7$, suggest that the thresholds 14 and 6 are blocked by ordinary group products rather than sporadic exceptions, so any higher threshold would need to exclude these families by an additional hypothesis.

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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

Here's my take. The paper extends known results on independence number of the comaximal subgroup graph in a meaningful way: α≤51 implies solvability, improving the earlier α≤8, and the supersolvability/nilpotency thresholds with explicit exceptions are new. The reduction to minimal simple groups via Thompson's classification is standard, and the counting arguments in Theorem 2.2 show genuine combinatorial skill. The sharpness examples are nice.

The problem is Theorem 2.1. After reducing to minimal simple groups, the proof eliminates four infinite families and PSL(3,3) with the single sentence 'One can check using SAGE or otherwise that in each of the cases α(Γ(G))≥52.' No code, no data, no analytic bound, no monotonicity argument that would reduce to a finite check. That is not a finite computation; the assertion covers infinitely many groups. If it fails for one prime, the theorem collapses. I agree with the stress-test note that this is load-bearing.

The same pattern appears elsewhere: Theorem 3.1 and 4.1 rely on 'exhaustive search' over small orders with no scripts, versions, or outputs, and Theorem 2.2 has steps like 'using this fact, it can be shown' that are not fully spelled out. On top of that, the paper uses the authors' own preprint [7] (arXiv:2403.01262) as a black box for Sub(M) bounds; self-citation isn't automatically a flaw, but here that preprint is unpublished and its results are invoked without proof, so the current paper isn't self-contained.

None of this makes me think the results are false. The thresholds are plausible and the techniques are consistent with what's known. But as written, the main theorem is not proven, and the computational claims are not reproducible. The paper deserves a serious referee, not because it's ready, but because the open problem and the sharp bounds are worth checking carefully. I'd send it to peer review with a request for major revision: supply a proof or verifiable code for the infinite family, include full computational data, and either prove or cite a published version of the results in [7].

Recommendation: engage with it, but only with the expectation that the gaps get closed.

Referee Report

4 major / 4 minor

Summary. This paper studies the comaximal subgroup graph Γ(G) of a finite group G and claims sharp thresholds on the independence number α(Γ(G)) that force solvability, supersolvability, and nilpotency. Theorem 2.1 asserts that α(Γ(G))≤51 implies G is solvable, with A5 as a sharpness example (α(Γ(A5))=52); Theorem 2.2 asserts that Γ(G)≅Γ(A5) implies G≅A5. Theorem 3.1 asserts that α≤14 implies supersolvability except for A4, SL(2,3), and (Z2×Z2)⋊Z9, and Theorem 4.1 asserts that α≤6 implies nilpotency except for five explicit groups. The proofs combine elementary subgroup-counting arguments with appeals to a preprint by the same authors, [7], and to SAGE/GAP exhaustive searches.

Significance. If the three thresholds are correct, they would be clean, sharp results that substantially improve the known α≤8 solvability criterion, and the graph-theoretic approach is natural. The paper is not purely existential: the main claims are concrete and falsifiable. However, the proof as written leaves the elimination of infinite families of minimal simple groups to a bare assertion and delegates key finite classifications to unreported searches, so the significance is conditional on filling those gaps.

major comments (4)

[Section 2, proof of Theorem 2.1] The sole step ruling out the infinite Thompson families is the sentence 'One can check using SAGE [14] or otherwise that in each of the cases α(Γ(G))≥52.' This is not a finite computation: the list contains PSL(2,2^p), PSL(2,3^p), PSL(2,p) with p≡±2 mod 5, and Sz(2^p) for infinitely many primes p. No uniform lower bound, no algorithm with a termination proof, and no monotonicity argument is provided. Since this assertion is the only step eliminating the minimal simple groups, Theorem 2.1 is not proved as written.
[Section 2, Theorems 2.1 and 2.2] Both theorems rely on results from the authors' arXiv preprint [7] as black boxes: Theorem 2.1 uses 'Theorem 2.1 [7]' to infer that Sub(M)≤52 implies M is solvable, and Theorem 2.2 ends with 'by Theorem 2.2 [7], G≅A5' from Sub(G)=59. These external results are not stated, and the paper's main conclusions are not verifiable without them. The authors should either include the statements and proofs, or supply a published reference.
[Sections 3 and 4] The proofs of Theorems 3.1 and 4.1 depend on a series of exhaustive searches over groups of orders 56, 18, 36, 48, 12, 24, 72, 216, 144, 96, and the reduced orders in Theorem 4.1, with no scripts, version numbers, or outputs. For example, the exclusion of non-supersolvable groups of order 56 ('by checking all non-supersolvable groups of order 56, we see...') and the final classification in Theorem 4.1 ('an exhaustive search ... revealed that only groups ... are ...') are exactly the content of the relevant step. These claims are not reproducible and need to be replaced by supplied code with documentation, or by analytic arguments.
[Section 2, proof of Theorem 2.2] The proof contains several essential steps that are merely asserted: 'Using this fact, it can be shown that G has unique subgroups of order 2, 2^2, ..., 2^{β−3}', and later that there are at most five subgroups of order 2^{β−2} and 2^{β−1}. The latter bound is used to derive the contradiction that closes Claim 2, but no argument is given for why five is the maximum or why the exceptional case cannot occur. These gaps, together with the unstated Theorem 2.2 [7], leave Theorem 2.2 incomplete.

minor comments (4)

[Remark 2.1] The sharpness value α(Γ(A5))=52 is asserted without computation; a short derivation from the subgroup lattice of A5 would make the claimed optimality checkable.
[Theorem 2.2, first paragraph] The description of Γ(A5) as K_{5,12} together with 40 isolated vertices is asserted; a sentence indicating which subgroups form the partite sets would be helpful.
[Data Availability statement] The statement that 'no datasets were generated or analysed' is inconsistent with the many exhaustive searches described in the proofs; if computations were used, the scripts and outputs should be listed there.
[Reference [14]] The SAGE reference is to version 7.3 (2016); the searches should identify the actual version and packages used. Also, the GAP searches mentioned in Theorem 2.2 are not referenced at all.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the independence-number thresholds are derived from subgroup-count results and finite computations, though the infinite Thompson-family check in Theorem 2.1 is unproved as written.

full rationale

The paper never defines one of its key quantities in terms of the conclusion, and no equation or definitional identity reduces the claims to their inputs. The independence-number bound α(Γ(G))≤51 is translated into a subgroup-count bound on each maximal subgroup via the elementary observation that a maximal subgroup together with its proper nontrivial subgroups forms an independent set. At that point the proof invokes reference [7], a separate preprint by the same authors, for parameter-free statements about the number of subgroups: Sub(M)≤52 implies M is solvable, and non-solvable with Sub(G)=59 implies G≅A5. These citations are load-bearing, but they are not circular in the forbidden sense: the current target is about α(Γ(G)), while [7] concerns Sub(G), and the assumptions of [7] do not include the present graph-theoretic conclusion. The same pattern holds in Theorems 3.1 and 4.1, where [7] supplies subgroup-count classification results and the remaining work uses finite exhaustive searches in SAGE/GAP that are reproducible in principle. The only serious weakness is in the proof of Theorem 2.1: the sentence 'One can check using SAGE [14] or otherwise that in each of the cases α(Γ(G))≥52' is asserted for four infinite Thompson families without a uniform analytic bound, a termination argument, or reproducible data. That is a proof gap and a correctness risk, but it is not circularity: it is an external computational claim about specific simple groups, not a fitted parameter, a renamed conclusion, or a self-referential definition. Accordingly, no circular step is present, and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper rests on standard group-theoretic tools plus two categories of unverified inputs: the same authors' prior preprint [7] and multiple computational assertions (finite and infinite) that lack reproducible artifacts. No new entities or fitted parameters are introduced.

assumptions (9)

domain assumption Theorem 2.1 of [7] (same authors' preprint): if Sub(G)≤52 then G is solvable; variants with Sub(M)≤7, Sub(M)<13, and the non-solvable Sub(G)=59 implies G≅A5
Used in Theorems 2.1, 2.2, 3.1, and 4.1 to convert bounds on number of subgroups into solvability, supersolvability, nilpotency, or isomorphism to A5. This source is an unpublished arXiv preprint by the same authors and is not independently verified here.
standard math Thompson's classification of finite minimal simple groups
Invoked in Theorem 2.1 to list possible minimal simple groups after reducing to the Frattini quotient.
standard math Hall's theorem for solvable groups (existence of Hall complements)
Used in Theorem 2.2 to produce Sylow complements in groups with three or more prime factors.
standard math Frattini argument
Used repeatedly in Theorem 2.2, in Claims 3 and 4, to control normalizers of Sylow subgroups.
standard math Theorem 9.1.9 of Robinson [12] on minimal non-nilpotent groups
Used in Theorem 4.1 to reduce the nilpotency problem to groups of order p^m q^n with a unique Sylow p-subgroup.
standard math Ore's theorem (Proposition 1.3): two maximal subgroups of a finite solvable group generate G or are conjugate
Used in Theorem 2.2, Claim 2, and in the counting arguments in Section 3.
standard math Aivazidis-Muller Proposition 1.1 on minimal number of subgroups of non-cyclic p-groups
Used to lower-bound the number of subgroups in p-groups, for example in Theorem 4.1 when counting subgroups of P.
ad hoc to paper Asserted finite computational results: no non-supersolvable group of orders 56, 18, 36, 48, 24, 72, 216, 144, 96 satisfies the respective condition, and no non-nilpotent group of the reduced orders satisfies α≤6 except the five listed
Stated as exhaustive search in GAP or SAGE without scripts, versions, or outputs. These computations are load-bearing for the exception lists in Theorems 3.1 and 4.1.
ad hoc to paper Infinite-family assertion: α(Γ(G))≥52 for every minimal simple group in Thompson's list
Stated as 'One can check using SAGE or otherwise' without proof or artifact. This is the weakest assumption in the paper, and it is essential for Theorem 2.1.

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Cite this review

Pith. "Pith review of On Independence Number of Comaximal Subgroup Graph." pith.science (2026). https://pith.science/paper/VEVETOCN

@misc{pith2026250603848,
  author       = {Pith},
  title        = {Pith review of: On Independence Number of Comaximal Subgroup Graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VEVETOCN}},
  note         = {Machine review of arXiv:2506.03848}
}

abstract

In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph $\Gamma(G)$ that guarantee solvability, supersolvability, and nilpotency of the underlying group $G$. Specifically: \begin{itemize} \item For solvability, we prove that any group $G$ with independence number $\alpha(\Gamma(G))\leq 51$ must be solvable, and show that the alternating group $A_5$ is uniquely determined by its graph. \item For supersolvability, we show that $\alpha(\Gamma(G))\leq 14$ implies $G$ is supersolvable, except for three explicit exceptions. \item For nilpotency, we prove that $\alpha(\Gamma(G))\leq 6$ ensures nilpotency, except for five groups. \end{itemize} Finally, we conclude with some open issues involving domination parameters.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [7]

Das and A

A. Das and A. Mandal, Solvability of a group based on its number of subgroups, https://arxiv.org/pdf/2403.01262

work page arXiv

[2] [14]

Stein and others: Sage Mathematics Software (Version 7.3), Release Date: 04.08.2016,http://www.sagemath.org

W. Stein and others: Sage Mathematics Software (Version 7.3), Release Date: 04.08.2016,http://www.sagemath.org

work page 2016

[3] [1]

Aivazidis and T

S. Aivazidis and T. Muller, Finite non-cyclicp-groups whose number of subgroups is minimal, Archiv der Mathematik, 114, pp 13-17,2020

work page 2020

[4] [2]

Akbari, B

S. Akbari, B. Miraftab and R. Nikandish, Co-maximal Graphs of Subgroups of Groups, Canadian Math Bulletin, Vol. 60(1), pp.12-25, 2017

work page 2017

[5] [3]

Cameron, Graphs defined on groups, International Journal of Group Theory, Vol

P.J. Cameron, Graphs defined on groups, International Journal of Group Theory, Vol. 11, Issue 2, Sl. 2, pp. 53-107, 2022

work page 2022

[6] [4]

A. Das, M. Saha and S. Alkaseasbeh, On Co-Maximal Subgroup Graph of a Group, Ricerche Di Mathematica, Vol. 73, pp. 2075-2089, 2024

work page 2024

[7] [5]

Das and M

A. Das and M. Saha, On Co-Maximal Subgroup Graph of a Group - II, Ricerche Di Mathematica, Vol. 74, pp. 91-104, 2025

work page 2025

[8] [6]

Das and M

A. Das and M. Saha, On Co-Maximal Subgroup Graph of Dn, Communications in Combinatorics and Optimization, Vol. 10, Issue 3, pp. 701-715, 2025

work page 2025