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arxiv: 2607.06434 · v1 · pith:EHVVU6OG · submitted 2026-07-07 · math.GR

On two questions from the Kourovka Notebook concerning maximal subgroups

Antonio Beltr\'an , Fernando Viudez This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 05:41 UTCglm-5.2pith:EHVVU6OGrecord.jsonopen to challenge →

classification math.GR MSC 20E2820D1020D0620D08
keywords finite groupsmaximal subgroupsp-nilpotentp-decomposableKourovka Notebooknon-abelian simple groupsalternating groupssporadic groups
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The pith

Odd-prime maximal subgroup conditions force finite groups into p-nilpotence

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

This paper settles two open questions from the Kourovka Notebook about finite groups whose maximal subgroups are each either non-abelian simple or p-nilpotent (respectively, p-decomposable), for a fixed odd prime p. The authors prove that any non-simple, non-solvable group satisfying this condition must itself be p-nilpotent (respectively, p-decomposable). The only way such a group can also possess a non-abelian simple maximal subgroup M is if the whole group is M × C_p or M ⋊ C_p, where M is a non-abelian simple group whose order is not divisible by p. This means the condition, which initially appears to permit rich structure, actually collapses almost entirely: the non-abelian composition factors are precisely the non-abelian simple p'-groups. The paper also classifies which alternating groups and sporadic simple groups satisfy the condition for specific primes, and for p = 2 gives a complete classification in terms of certain PSL_2(q), PGL_2(q), and related groups.

Core claim

The central mechanism is a reduction argument: given a non-simple, non-solvable group G satisfying the maximal-subgroup condition, the authors take a minimal normal subgroup N and split into two cases depending on whether N lies inside the non-abelian simple maximal subgroup M. In one case, they show G decomposes as M × C_p; in the other, they reduce to showing every maximal subgroup other than M is either normal or p-nilpotent, then invoke a prior result (based on the Glauberman-Thompson criterion) that such a group is p-solvable, which forces the quotient G/M to be C_p. The condition is so restrictive that it forces the entire group into p-nilpotence, leaving only the trivial extension byC

What carries the argument

The proofs of Theorems A and B avoid the Classification of Finite Simple Groups, relying instead on Itô's theorem on minimal non-p-nilpotent groups and a p-solvability criterion derived from the Glauberman-Thompson p-nilpotence theorem. Theorems C and D use the Liebeck–Praeger–Saxl classification of maximal subgroups of alternating groups and the Atlas of Finite Groups for sporadic groups. Theorem E uses the known classification of maximal subgroups of PSL_2(q).

If this is right

  • The composition factors of any non-simple group satisfying the maximal-subgroup condition for odd p are exactly the non-abelian simple p'-groups, giving a clean answer to Kourovka Problems 19.57 and 19.58.
  • For p = 2, the paper shows every group satisfying the 2-decomposable analogue is solvable (Corollary F), a result that does not explicitly appear in the prior literature.
  • The complete classification for p = 2 (Theorem E) refines the earlier result of Monakhov and Tyutyanov by pinning down exactly which PSL_2(q), PGL_2(q), and extensions satisfy the condition.
  • The case of simple groups of Lie type for odd p remains open; the paper provides scattered examples but no general classification, suggesting this as the natural next frontier.

Where Pith is reading between the lines

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

  • The collapse to p-nilpotence suggests that the condition 'every maximal subgroup is non-abelian simple or p-nilpotent' is, for non-simple groups, essentially a disguised form of p-nilpotence itself — the non-abelian simple maximal subgroup, if it exists, is structurally forced to be a p'-group acting trivially or nearly trivially.
  • A natural extension would be to replace 'p-nilpotent' with other local properties (e.g., p-supersolvable, p-closed) and ask whether a similar collapse occurs; the proof technique of reducing to normal-or-p-nilpotent and invoking p-solvability criteria might generalize.
  • The difficulty of extending Theorems C and D to groups of Lie type suggests that the maximal subgroup structure of these groups is too rich for the condition to hold except in exceptional arithmetic circumstances, which could be tested computationally for small-rank families.

Load-bearing premise

The proof of Theorem A's second case relies on an external result stating that a finite group whose maximal subgroups are each either normal or p-nilpotent is p-solvable for odd p; if that result had hidden hypotheses or gaps, this case of the argument would fail. Theorems C and D also depend on the completeness of published classifications of maximal subgroups of alternating and sporadic groups.

What would settle it

If one could exhibit a non-simple, non-solvable finite group that is not p-nilpotent but in which every maximal subgroup is either non-abelian simple or p-nilpotent for some odd p, Theorem A would be false.

read the original abstract

Let \(p\) be a prime number. When \(p\) is odd, we study finite groups in which every maximal subgroup is either non-abelian simple or \(p\)-nilpotent, as well as those in which every maximal subgroup is either non-abelian simple or \(p\)-decomposable. We prove that every non-simple, non-solvable group satisfying these criteria is \(p\)-nilpotent, and \(p\)-decomposable, respectively. This answers two open questions posed by V.S. Monakhov and I.N. Tyutyanov in the Kourovka Notebook. Additionally, if \(p=2\), we improve the main result of Monakhov and Tyutyanov by providing a complete classification of non-solvable groups whose maximal subgroups are either non-abelian simple or \(2\)-nilpotent.

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.

Referee Report

0 major / 6 minor

Summary. The paper answers Kourovka Notebook Problems 19.57 and 19.58, which ask for the non-abelian composition factors of finite groups whose maximal subgroups are either non-abelian simple or p-nilpotent (resp. p-decomposable), for odd primes p. Theorem A proves that any non-simple, non-solvable group satisfying (∗_p) is p-nilpotent, with an explicit structure theorem when such a group has a non-abelian simple maximal subgroup. Theorem B establishes the analogous result for p-decomposability. Theorems C and D determine exactly which alternating and sporadic simple groups satisfy (∗_p) or (∗_{p-p'}). Theorem E provides a complete classification for p = 2, refining the earlier result of Monakhov–Tyutyanov [18], and Corollary F shows that every group satisfying (∗_{2-2'}) is solvable. The proofs of Theorems A and B are CFSG-free, relying instead on the Glauberman–Thompson criterion via [1, Corollary 4.2] and Itô's theorem on minimal non-p-nilpotent groups.

Significance. The paper resolves two stated open problems from the Kourovka Notebook, which is a significant contribution. A notable strength is that the core results (Theorems A and B) are proved without the Classification of Finite Simple Groups, making them independent of that machinery. Theorem E provides a complete and explicit classification for p = 2, going substantially beyond [18, Theorem 2] by fully characterizing the group structure rather than only constraining composition factors. The case analyses in Theorems C and D, while necessarily computational, are thoroughly documented with references to the Atlas, the Liebeck–Praeger–Saxl classification, and subsequent complete classifications of maximal subgroups of sporadic groups. The arithmetic verifications (e.g., the congruence reduction in Theorem E(1)(a)) are correct and checkable.

minor comments (6)
  1. Abstract: 'the main result on [18]' should read 'of [18]'.
  2. Theorem E(2): the statement 'q = p^{2a}, a ≥ 0' is ambiguous when a = 0, since p^{2·0} = p^0 = 1 is not a valid argument for PGL_2. The proof clarifies that a = 0 corresponds to q = p (a prime), but the theorem statement should make this explicit, e.g., by stating the a = 0 case separately or by writing 'q = p^{2a} for a ≥ 1, or q = p for a = 0'.
  3. Proof of Theorem E, Case 2 (p. 13): the formula '|Out(T)| = (2, q−1)2a = 2a+1' appears to have a formatting issue. The intended formula is |Out(PSL_2(p^{2a}))| = (2, q−1) · 2a = 2 · 2a = 2^{a+1}, not 2a+1. Please verify the superscript rendering.
  4. Proof of Theorem C, Case 2, p = 17 (p. 8): the maximality of PSL_2(16):C_4 in A_17 is verified using GAP [5]. If a published reference for this containment is available, it would strengthen the argument, though the computational verification is acceptable.
  5. Introduction (p. 2): the fact that Theorems A and B do not require CFSG is mentioned only at the start of Section 2. Stating this in the Introduction would be informative for readers assessing the foundational dependencies of the main results.
  6. Proof of Theorem D: the case-by-case analysis is necessarily terse. For a few of the larger sporadic groups (e.g., cases 25–26 for B and M), it would help the reader to briefly indicate which maximal subgroup reference was used for each prime, though the current citations to [20] and [4] are sufficient for verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment. We are grateful for the referee's recognition of the paper's contributions, particularly the CFSG-free nature of the proofs of Theorems A and B, and the complete classification in Theorem E for p = 2. We address the referee's comments below.

read point-by-point responses

  1. Referee: REFEREE RECOMMENDATION: accept

    Authors: We thank the referee for recommending acceptance. We are pleased that the referee finds the paper's resolution of Kourovka Notebook Problems 19.57 and 19.58 to be a significant contribution, and that the referee has verified the correctness of the arithmetic computations and case analyses. revision: no

  2. Referee: MAJOR COMMENTS: (none listed)

    Authors: The referee did not raise any major comments requiring revision. We note one minor typographical issue in the manuscript that we will correct in the final version: in the sentence preceding Section 3, the phrase 'subsequently he analyzes' should read 'subsequently the analysis of' to fix a grammatical error. This is a purely editorial correction that does not affect any mathematical content. revision: yes

Circularity Check

0 steps flagged

No significant circularity found; the derivation is self-contained against external benchmarks.

full rationale

The paper's central results (Theorems A–F) are derived from standard group-theoretic arguments using externally verifiable classifications (Atlas, Liebeck–Praeger–Saxl, Bray–Holt–Roney-Dougal) and standard theorems (Itô's theorem, Glauberman–Thompson). The two self-citations ([1] Beltrán–Shao and [19] Shao–Beltrán) are invoked as load-bearing lemmas, but they state genuinely different results from the present paper's claims: [1, Corollary 4.2] states that groups whose maximal subgroups are normal or p-nilpotent are p-solvable (a different hypothesis from the paper's condition (∗_p)), and [19, Theorem A] concerns groups whose maximal subgroups are 2-nilpotent or normal. Neither cited result is equivalent to the paper's theorems by construction. The internal logic of Theorem A (Case 2) correctly reduces to the hypotheses of [1, Corollary 4.2] by showing every maximal subgroup distinct from M is p-nilpotent and M is normal, then invokes the corollary to get p-solvability, which independently forces q=p. Theorem B's derivation from Theorem A uses a standard coprime action argument. Theorems C–D are case-by-case verifications against external classification databases. Theorem E relies on [18, Theorem 2] (Monakhov–Tyutyanov, not self-cited) and [6] (Giudici, not self-cited). The self-citations are normal scholarly practice and do not create circular dependencies. The minor score of 2 reflects the presence of self-citation at load-bearing points, but the cited results have independent content and are not equivalent to the paper's conclusions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 0 invented entities

No new entities are postulated. This is a classification theorem within established finite group theory.

axioms (8)
  • standard math Itô's theorem: a minimal non-p-nilpotent group is solvable [7, Theorem IV.5.4]
    Used in proofs of Theorems A and E to derive contradictions when a group would be minimal non-p-nilpotent. Classical result.
  • domain assumption [1, Corollary 4.2]: a finite group whose maximal subgroups are either normal or p-nilpotent is p-solvable for odd p
    Load-bearing in Theorem A, Case 2 (p. 4). Co-authored by the first author of this paper, but is a different statement from the paper's claims.
  • standard math Classification of maximal subgroups of alternating groups (Liebeck–Praeger–Saxl [14])
    Used throughout Theorem C to determine which maximal subgroups of A_n exist.
  • domain assumption Completeness of maximal subgroup classifications for sporadic groups from the Atlas [3] and subsequent references
    Theorem D's correctness depends entirely on these external classifications being complete and accurate.
  • domain assumption [18, Theorem 2]: groups satisfying (∗_2) have at most one non-abelian composition factor, isomorphic to PSL_2(q)
    Starting point for Theorem E's proof (p. 13). Published result by Monakhov–Tyutyanov.
  • domain assumption [19, Theorem A]: classification of groups whose maximal subgroups are 2-nilpotent or normal
    Used in Theorem E, Case 2 (p. 13) to constrain the structure of T. Co-authored by the first author.
  • standard math Theorem 3.1 (Jones): primitive permutation groups containing a cycle with k fixed points
    Used in Theorem C to classify primitive subgroups of A_n containing p-cycles.
  • standard math Theorem 4.1: classification of maximal subgroups of PSL_2(q) from Bray–Holt–Roney-Dougal [2]
    Used in Theorem E to determine when PSL_2(q) satisfies (∗_2).

pith-pipeline@v1.1.0-glm · 19524 in / 3657 out tokens · 210723 ms · 2026-07-08T05:41:22.194410+00:00 · methodology

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

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