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On a generalisation of Cameron's base size conjecture

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Pith's one-line read Seven conjugates always suffice for non-standard subgroup tuples

desk verdict Proves the regularity-number analogue of Cameron's conjecture (R_ns(G) ≤ 7, equality iff M24) with new Lie type bounds; substantial and probably correct, but the computational layer needs to be released and the E8(q) coverage citation checked. read the letter →

arxiv 2511.08705 v4 pith:5IM3NDGV submitted 2025-11-11 math.GR

Marina Anagnostopoulou-Merkouri This is my paper · ORCID
classification math.GR MSC 20B0520B1520D06
keywords basesizeregularitynumbernon-standardmaximalsubgroupsfixedpointratiosalmostsimplegroupsofLietypeprobabilisticgrouptheorysporadicM24
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 paper proves that a single absolute bound governs how many subgroups of a finite almost simple group must be conjugated before they can be made to meet trivially, even when the subgroups are chosen independently rather than as repeats of one stabilizer. The main theorem states that any seven non-standard maximal subgroups can be simultaneously conjugated to have trivial intersection, and that seven is needed only by the sporadic group M24 with its M23 stabilizer. For classical groups five conjugates suffice, with the lone exception U6(2).2; for exceptional groups of Lie type, any six proper subgroups suffice. On the way the paper establishes quantitative probabilistic estimates—fixed-point-ratio bounds and zeta-type function evaluations—that are likely to be reused independently.

What carries the argument

The engine is the probabilistic method: for a tuple τ, the probability that a random point lies outside all regular orbits is bounded above by Σ_i |x_i^G| · Π_j fpr(x_i, G/H_j), where x_i runs over prime-order conjugacy classes and fpr denotes the fixed point ratio. The proof reduces to worst-case fixed-point-ratio estimates over all non-standard subgroups together with the zeta-type function η_G(t) = Σ_C |C|^{-t} over prime-order G0-classes; showing η_G(t) < 1 for a suitable t forces the tuple to be regular. Low-dimensional classical groups require sharpened fixed-point-ratio bounds derived from maximal-subgroup data and conjugacy-class tables; high-dimensional classical groups are handled

What would settle it

Produce a maximal subgroup of E8(q) with order larger than q^88 that is not among the families cited in the paper, and exhibit a prime-order element whose fixed-point proportion exceeds the recorded bounds; this would collapse the uniform estimate behind the exceptional-group theorem.

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Extended reading notes

Core claim

The central discovery is that the base-size bound seven survives a substantial weakening of the old conjectures: the tuple of subgroups is no longer required to be conjugate. Defining the regularity number R_ns(G) as the least k such that every k-tuple of non-standard core-free maximal subgroups is regular—that is, admits an orbit with trivial pointwise stabilizer on the product of coset spaces—the paper shows R_ns(G) ≤ 7 for every finite almost simple group, with equality exactly for G = M24. The classical-group case is settled more sharply at R_ns(G) ≤ 5, with equality iff G = U6(2).2, and the exceptional-group case at R(G) ≤ 6, meaning in particular that six arbitrary proper subgroups of

Load-bearing premise

The load-bearing premise is that, in the unresolved E8(q) case, the worst-case fixed point ratio is achieved by a maximal subgroup from the already classified families; the full classification of E8(q) maximal subgroups is still open.

Editorial extensions

If this is right

  • For every finite almost simple group, any seven non-standard maximal subgroups can be conjugated so that their intersection is trivial; only M24 with M23 stabilizers genuinely requires seven.
  • In classical groups, five conjugates always suffice, with the single exception U6(2).2, whose unique non-regular 4-tuple is four copies of U4(3).2².
  • In every simple exceptional group of Lie type, any six proper subgroups can be simultaneously conjugated to have trivial intersection.
  • As |G| grows, the probability that a randomly chosen 6-tuple of non-standard subgroups is regular tends to 1.
  • The original single-action conjecture is recovered as the special case where all subgroups in the tuple are pairwise conjugate.

Reading between the lines

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

  • The domination argument—any core-free subgroup sits inside a maximal one—suggests that the 7-tuple statement may extend to arbitrary non-standard core-free subgroups, not only maximal ones, although the paper only formulates the maximal version.
  • The refined zeta-type bounds established here, such as η_G(1/5) < 0.9 for PSp6(q) and Ω7(q), are standalone quantitative estimates that could be reused in spectral or random-walk questions on finite groups of Lie type.
  • A direct computational check of every 6-tuple among the known maximal subgroups of E8(2) would test the sharpest finite case of the exceptional-group theorem; a non-regular 6-tuple there would refute it, while a positive check would still leave the infinite family dependent on the open classification.
  • If a future classification of the missing E8(q) maximal subgroups produces a previously unknown family, the fixed-point-ratio bounds behind the exceptional-group theorem would need to be re-verified against that family—a concrete revision point that would not necessarily change the main theorem.
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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

2 major / 4 minor

Summary. The paper proves a generalisation of Cameron's base size conjecture to heterogeneous tuples of subgroups. For a finite almost simple group G, it defines R_ns(G) as the least r such that every r-tuple of non-standard maximal subgroups admits a regular orbit on the product of coset spaces, and proves Theorem 2: R_ns(G) ≤ 7, with equality if and only if G = M24. The proof splits into classical and exceptional Lie type cases. For classical groups it proves the stronger Theorem 3.1: R_ns(G) ≤ 5, with equality only for U6(2).2, and P(G,4)→1 as |G|→∞. For exceptional groups it proves Theorem 7.1: every 6-tuple of core-free subgroups of an almost simple exceptional group is regular, with Q(G,τ)→0 as q→∞. The methods are probabilistic, based on fixed point ratio estimates, with extensive use of published classification results, known fixed point ratio bounds, and computational checks in GAP/Magma.

Significance. If correct, this settles a natural and well-motivated generalisation of Cameron's conjecture and gives a remarkably strong uniform bound for non-conjugate tuples. The paper contains useful standalone contributions: the fixed point ratio bounds in Theorem 4.1 and Tables 13–17, the eta-function estimates in Section 5, and the refined asymptotic statement P(G,4),P(G,6)→1. There is no circularity: the target theorem is not used as an input, the constants in the bounds are fixed and derived, and the small-base-size results of [18] are used only as comparisons or for special tuples. However, the E8(q) case rests on an external structural claim about large non-parabolic maximal subgroups that is not reconciled with the paper's own statement that the S-collection of E8(q) is unclassified, and several essential computations rely on private or unshipped data. The classical part and the E1/E2 exceptional parts appear sound; the E8 gap is load-bearing and needs to be closed or made explicit.

major comments (2)
  1. [§7.1, Prop 10.10 (E8(q))] For E8(q), the non-parabolic case splits at |H|≤q^88. For |H|>q^88, the proof says: 'The possibilities for H are described in [26, 4.2]' and fpr(x,G/H)≤c_i follows by inspecting [26, 4.4–4.6]. This is load-bearing: the smallest classes in L2 have size about q^125, so an unlisted maximal subgroup of order q^90 would give fpr≈q^{-35} > q^{-37}=c_2. Yet §7.1 states that the S-collection of E8(q) maximal subgroups is not classified even up to isomorphism, and the paper gives no argument that every unclassified member has order ≤q^88. The sentence 'The current state of the art regarding the groups in class S will not be relevant to us' is therefore not justified by the text. Please supply a proof or a precise reference for the coverage claim for maximal subgroups of order >q^88, or restrict/restructure the argument.
  2. [Sections 2.2, 5, 6, 9, 10] Several essential steps rest on computations that cannot be audited from the manuscript. The Foulkes-function data [65] is a private communication, and numerous 'one can check' assertions (e.g., Propositions 5.1–5.3, 6.3–6.5, 10.10) are not accompanied by code or output. The Magma/GAP fusion computations in Propositions 8.2 and 8.4 are described only in outline. These computations cover the small-q cases and the parabolic fixed-point-ratio estimates that are needed for the exceptional-group proof. Please deposit the code and/or output, or provide enough detail (inputs, intermediate data, verification scripts) for an independent reader to reproduce the calculations.
minor comments (4)
  1. [Throughout] Typos: 'GENERALISA TION' in the title; 'permuatation' in Lemma 2.3; 'the proofs of the bounds in the proofs of' in Proposition 4.12; 'it is left to check' in Proposition 4.4. A careful proofreading pass is needed.
  2. [Theorem 7.1] Theorem 7.1 is stated for 6-tuples of core-free subgroups, but the fixed point ratio estimates in the proof are stated for maximal subgroups. It would be helpful to add the standard observation that if H≤M, then fpr(x,G/H)≤fpr(x,G/M), so one may pass to a maximal overgroup; this is implicit but should be explicit.
  3. [Table 7] Several β-values are given as decimals (0.4, 0.225, 0.32). Exact rationals would be preferable, especially since the proofs refer to inequalities such as log|x^{G0}∩H|/log|x^{G0}|<0.68.
  4. [Section 4] Many cases in Propositions 4.3–4.6 and 4.10–4.16 are left 'for the reader to check'. This is acceptable in a research paper, but the overall verification burden is large; a supplementary file with the omitted computations would greatly improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lie-type proof reduces to external fixed-point-ratio estimates and direct computations, not to its own conclusion.

full rationale

Walked the derivation chain. Theorem 2 splits into alternating/sporadic (via [2]) and Lie type; the Lie-type part is proved as Theorem 3.1 for classical groups and Theorem 7.1 for exceptional groups. The classical proof bounds \hat Q(G,\tau) using Burness's fpr estimates [14-17], the new estimates in Theorem 4.1, and \eta_G bounds in Section 5, with small-q cases checked in Magma; no parameter is fitted to the target R_ns(G)\le5. The exceptional proof sums |x^G| f(x,q)^6 using fpr bounds from [52], [26], [19], Luebeck's Foulkes data [65], and explicit parabolic fixed-point computations (Prop 9.1, Cor 9.2); the target is not an input. Same-author citations [1,2] are published/independent prior regularity results for alternating/sporadic groups and small computations, and they do not assume the Lie-type claim, so they are not load-bearing self-citation in the circularity sense. Several 'one can check' steps (e.g., Propositions 4.3-4.16, 10.10) are computational omissions, not circular reductions. I weighed the explicit limitation in Section 7.1 ('It is currently an open problem to determine the members of S even up to isomorphism') against the coverage assertion in Prop 10.10 that for non-parabolic H with |H|>q^88 'the possibilities for H are described in [26, 4.2]'. This is a substantive external coverage/correctness risk, but it is not a reduction of the theorem to its own input or to a fitted parameter, so it does not constitute circularity under the rules.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the CFSG and the published subgroup-structure/fpr literature (Aschbacher, Kleidman–Liebeck, Bray–Holt–Roney-Dougal, Liebeck–Seitz, Craven, Lawther–Liebeck–Seitz, Burness), plus a computational layer that is not fully public. There are no empirical free parameters and no invented entities; the hand-chosen constants are verification-level bounds, not fitted to the target result.

free parameters (2)
  • beta exponents in Theorem 4.1 / Table 7 (1/3, 0.4, 0.225, 0.32, 9/20, 3/10)
    Hand-chosen exponents in fixed point ratio bounds fpr(x,G/H) < |x^{G0}|^{-beta} for low-dimensional classical groups; selected because generic bounds fail (Remark 4.7 shows 1/3 cannot hold for PSL6(q) with C_G(x) = GL3(q)^2, forcing beta = 0.32). Each is validated by direct log-ratio computation; they are proof-design constants, not fitted to the conclusion.
  • g(x,q) bounds in Tables 13–17 (exceptional non-parabolic actions)
    Recorded worst-case fpr upper bounds of the form c·q^{-d} for E6^epsilon(q), E7(q), F4(q); derived from [26] and [52] with case-by-case improvements in Section 8. Constants chosen ad hoc to make the final bQ < 1 sums converge and verified numerically for small q; verification-level, not fitted in a load-bearing sense.
assumptions (5)
  • standard math Classification of Finite Simple Groups and Aschbacher's subgroup structure theorem (Thm 3.3)
    Background for the structure of almost simple groups, their automorphisms, and the maximal subgroup collections C1..C8, S, N used throughout Sections 3–10.
  • domain assumption Maximal subgroup classifications for exceptional groups (Liebeck–Seitz, Craven, Malle, Kleidman, Wilson)
    Section 7.1: for E7(q) and E8(q) the S-collection is not fully classified; the proof assumes the sufficiently large subgroups are covered by [26, 4.2] (see Prop 10.10).
  • domain assumption Lawther–Liebeck–Seitz fixed point ratio theorems and Burness's classical fpr theorems [14–17] used as black boxes
    Sections 3.3, 4, 8: the paper improves some of these bounds but relies on the original statements for the majority of cases.
  • standard math Lang–Steinberg theorem and Steinberg's conjugacy class theorems (also Proposition 3.5)
    Used in Sections 3.2 and 8 (e.g., equality of G0- and Inndiag-classes for semisimple elements; connectedness arguments for centralisers).
  • domain assumption Correctness of Magma/GAP computations and of privately communicated data (Luebeck's Foulkes functions [65])
    Sections 2.2, 9.2, 10: the boundedness of the eta and bQ expressions for all q in specified ranges is asserted on the basis of these computations; no public artifact or independent audit is provided.

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Pith. "Pith review of On a generalisation of Cameron's base size conjecture." pith.science (2026). https://pith.science/paper/5IM3NDGV

@misc{pith2026251108705,
  author       = {Pith},
  title        = {Pith review of: On a generalisation of Cameron's base size conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5IM3NDGV}},
  note         = {Machine review of arXiv:2511.08705}
}
abstract

Let $G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of $G$, denoted by $b(G, \Omega)$. Equivalently, $b(G, \Omega)$ is the minimal positive integer $k$ such that $G$ has a regular orbit on the Cartesian product $\Omega^k$. A well-known conjecture of Cameron from the 1990s asserts that if $G$ is an almost simple primitive group and $H$ is a so-called non-standard subgroup, then $b(G, \Omega) \leqslant 7$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action of degree $24$. This conjecture was settled in a series of papers by Burness et al. (2007-11). In this paper, we complete the proof of a natural generalisation of Cameron's conjecture. Our main result states that if $G$ is an almost simple group and $H_1, \ldots, H_k$ are any non-standard maximal subgroups of $G$ with $k \geqslant 7$, then $G$ has a regular orbit on $G/H_1 \times \cdots \times G/H_k$, noting that Cameron's original conjecture corresponds to the special case where the $H_i$ are pairwise conjugate subgroups. In addition, we show that the same conclusion holds with $k = 6$, unless $G = {\rm M}_{24}$ and each $H_i$ is isomorphic to ${\rm M}_{23}$. For example, this means that if $G$ is a simple exceptional group of Lie type and $H_1, \ldots, H_6$ are proper subgroups of $G$, then there exist elements $g_i \in G$ such that $\bigcap_i H_i^{g_i} = 1$. By applying recent work in a joint paper with Burness, we may assume $G$ is a group of Lie type and our proof uses probabilistic methods based on fixed point ratio estimates.

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