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Exceptional groups and the s-arc-transitivity of vertex-primitive digraphs, II

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Pith's one-line read This paper proves that every vertex-primitive s-arc-transitive digraph admitting an almost simple automorphism group with socle E7(q) or E8(q) has s ≤ 2, completing the boundedness question for all exceptional groups of Lie type.

desk verdict Last two exceptional socles handled, and the proof is probably right, but the E7(q) branch rests on an unpublished classification and unshipped Magma checks. read the letter →

arxiv 2607.14603 v2 pith:PHCNGFAS submitted 2026-07-16 math.GR

Lei Chen , Fu-Gang Yin This is my paper · ORCID
classification math.GR MSC 20B2520D0605C25
keywords s-arc-transitivedigraphvertex-primitivealmostsimplegroupexceptionalofLietypeE7(q)E8(q)maximalsubgroupshomogeneousfactorization
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

For digraphs, unlike graphs, the arc-transitivity parameter s can be arbitrarily large; the open question is whether vertex-primitive examples—digraphs whose automorphism group acts primitively on the vertex set—force s to be small. This paper settles the last open exceptional-group cases: if an almost simple group with socle E7(q) or E8(q) acts vertex-primitively on a connected s-arc-transitive digraph, then s ≤ 2, and any 2-arc-transitive example must have a vertex stabilizer taken from an explicit list. Together with earlier work on the other exceptional Lie-type groups, this answers the boundedness question for every exceptional group. A sympathetic reader would care because it sharply limits where highly symmetric primitive digraphs can exist.

What carries the argument

The load-bearing object is the homogeneous factorization condition G_v = G_uv G_vw—a factorization by two isomorphic subgroups—which 2-arc-transitivity imposes on the vertex stabilizer. The argument couples this with a reduction lemma bounding the 2-part of |L_v| by |L_uv|^3 times the order of the outer automorphism group, and uses primitive prime divisors (primes dividing q^n − 1 but no earlier q^i − 1) to isolate a normal simple factor. The arc-shift element g satisfies (u,v)^g = (v,w); Lemma 2.5 says g normalizes no proper nontrivial normal subgroup of G_v, which is what turns 'one factor contains a whole simple factor' into a contradiction. The proof is an exhaustive case analysis over t

What would settle it

Find a core-free maximal subgroup H of E7(q) or E8(q) outside the lists used in the paper that admits a homogeneous factorization H = AB with A ≅ B and |H : A| ≥ 3, where the element g with (u,v)^g = (v,w) normalizes no nontrivial normal subgroup of H; such a factorization would produce a G-vertex-primitive (G,2)-arc-transitive digraph and could lift to s ≥ 3. Equivalently, constructing any connected G-vertex-primitive (G,3)-arc-transitive digraph with socle E7(q) or E8(q) would refute Theorem 1.1.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: let G be an almost simple group with socle E7(q) or E8(q), and let Γ be a G-vertex-primitive (G,s)-arc-transitive digraph; then s ≤ 2. Moreover, if Γ is (G,2)-arc-transitive and the socle is any simple exceptional group of Lie type, the vertex stabilizer L_v must be one of the groups assembled in Table 6, ranging from small groups such as A6 and M12 to product and torus-type subgroups. The proof assumes s ≥ 2 and exploits the homogeneous factorization G_v = G_uv G_vw that 2-arc-transitivity forces, then runs through every core-free maximal subgroup of E7(q) and E8(q), organized into parabolic, maximal-rank, and remaining almost simple cases. In each family,

Load-bearing premise

The argument depends on the completeness of the maximal-subgroup classifications it cites—including an unpublished E7(q) list and one characteristic-3 E8(q) exceptional subgroup—and on the correctness of many computer-algebra factorization checks; if any maximal subgroup class were missing, a vertex stabilizer could exist that the proof never considers, and the s ≤ 2 conclusion would not follow.

Editorial extensions

If this is right

  • For every connected vertex-primitive digraph admitting an almost simple exceptional group of Lie type, s-arc-transitivity stops at s = 2; in particular, no such digraph is 3-arc-transitive.
  • Any 2-arc-transitive example with E7(q) or E8(q) socle has its vertex stabilizer in Table 6, so the search for actual examples is reduced to a finite list of stabilizer groups.
  • The boundedness question for vertex-primitive digraphs is answered affirmatively for all exceptional groups of Lie type when combined with earlier work.
  • Maximal parabolic vertex stabilizers are eliminated entirely for 2-arc-transitivity: in that case s ≤ 1.
  • Whether any 2-arc-transitive digraphs with E7(q) or E8(q) socle actually exist remains open; the result constrains but does not construct them.

Reading between the lines

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

  • A natural next test is to run the same factorization machinery against the remaining almost simple classical groups, where the boundedness question is not yet settled by this method; the E7/E8 proof supplies reusable prime-divisor arguments.
  • The explicit Table 6 stabilizer list suggests a targeted computational search: for each listed L_v, test whether a homogeneous factorization L_v = AB exists with index at least 3 and with the normalizer-freeness condition; existence would produce the first primitive 2-arc-transitive digraphs with E7/E8 socle, while nonexistence would strengthen the conclusion to full non-existence.
  • Because the proof leans on an unpublished E7 maximal-subgroup classification and on many computer checks without reproducible scripts, an independent verification of those inputs is the main thing that would change confidence in the theorem—though this is an editorial observation, not a flaw identified by the paper.
  • The parabolic-subgroup argument appears to be new and more direct than earlier approaches; it may apply to other Lie-type groups where parabolic cases were previously handled only by lengthy computation.
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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

3 major / 5 minor

Summary. The paper proves that if G is an almost simple group with socle E7(q) or E8(q) and Γ is a G-vertex-primitive (G,s)-arc-transitive digraph, then s≤2, and it lists the possible vertex stabilizers for (G,2)-arc-transitive examples. The main strategy, following the authors' companion paper, is to use the fact that s≥3 forces G_v=G_uvG_vw for a 2-arc u→v→w, and then to rule out, by a case analysis over the maximal subgroups of E7(q) and E8(q), every possible vertex stabilizer. Parabolic cases are handled by a Weyl-group double-coset argument combined with primitive prime divisors; maximal-rank cases use the Liebeck–Saxl–Seitz classification and a series of lemmas; the remaining cases depend on the classifications in [12], [10], [11], [18], together with numerous Magma computations. The paper also presents Table 6 listing the remaining possible vertex stabilizers across all exceptional groups.

Significance. If correct, Theorem 1.1 is a major result: it completes the solution of Giudici–Xia's boundedness question for all finite almost simple exceptional groups of Lie type, reducing the answer to the finite list in Table 6. The proof is not based on fitting parameters or circular use of the target theorem; it imports prior published lemmas from [31] and [32], uses Zsigmondy's theorem, and gives explicit number-theoretic contradictions in many cases. The explicit Table 6 is a useful falsifiable output. The main structural weakness is that the exhaustiveness of the case analysis rests on external classifications (including an unpublished preprint) and on dozens of unarchived Magma computations, so the proof cannot currently be independently audited. These are correctness-risk concerns rather than internal inconsistencies.

major comments (3)
  1. [§3.3.1] The unconditional statement of Theorem 1.1 depends on the completeness of the maximal-subgroup list for E7(q). The non-parabolic, non-maximal-rank cases are taken verbatim from [12], an unpublished 2022 arXiv preprint. The line “By [12] and [17]” is load-bearing: if [12] omits a maximal subgroup whose F*(L_v) is not almost simple (for instance a product-type subgroup), Lemma 2.10(a) does not apply, and such a subgroup could in principle admit the homogeneous factorization L_v=L_uvL_vw that would give s≥3. Since neither the contents of [12] nor a proof of the needed part is reproduced, the theorem as stated is not checkable from this manuscript. The authors should either use a published version of [12], supply a proof or a verifiable table of the maximal subgroups used, or make the theorem conditional on [12].
  2. [§3.1, Lemmas 3.6–3.27] The proof relies on many Magma computations, with no scripts, logs, or output files included. The computations decide nonexistence of factorisations and irreducibility of modules, which are exactly the steps that would break if a computational error occurred. Because this is a classification-style proof with an exhaustive case analysis, the computations need to be reproducible; otherwise the reader cannot distinguish a verified case from a black box. The concern is amplified by Remark 3.9, where a published structure in [20, Table 5.2] is corrected by a Magma computation: computational artefacts have already changed a group structure used in the proof.
  3. [§3.3.1, Lemma 3.22] The step from “S=PSL2(q)” to “we only need to consider PSL2(23)” is not argued. Lemma 2.10(a) rules out PSL2(q) only when the additional “almost simple” clause applies to G_v; for cases (vi) and (vii) this is not automatic from the displayed group structure. The subsequent discussion of R(G_v)=2 for q=23 shows that non-almost-simple vertex stabilizers are considered possible. The text should explain for which q the extension by Out(L) centralizes the PSL2(q) factor, why all other q are covered by the almost-simple clause, and why no other exceptional q arises. As written, this is a gap in a step that is necessary for the s≤2 conclusion.
minor comments (5)
  1. [Lemma 3.12] In the final paragraph, “Now, we deal with case (8)” should be “case (10)”; the lemma covers cases (9)–(12).
  2. [Lemma 3.23] Case (3) is titled “q is even and q is not a Mersenne prime,” but the paragraph immediately treats q=5 and q=9, which are odd. The title should say “q is odd and not a Mersenne prime” (or equivalent).
  3. [Lemmas 3.6 and 3.14] The expressions “with the index 24 or 23” and “with the index 24” should be “index 2^4 or 2^3” and “index 2^4” respectively; the current text is confusing.
  4. [Lemma 3.14] In the q=3 paragraph, “Now H=L” should presumably be “Now G=L” (since q=3 forces G=L); H is a small subgroup SL2(3)^4, not the whole group.
  5. [References] References [22], [23], and [30] appear in the bibliography but do not seem to be cited in the text; please check and either cite them or remove them.

Circularity Check

0 steps flagged · score 2.0 of 10

No substantiated circularity; the heavy reliance on external classifications and black-box Magma computations is a completeness/reproducibility risk, not circular reasoning.

full rationale

Walking the derivation chain, the target theorem s≤2 is obtained by assuming a (G,2)-arc-transitive digraph, translating 2-arc-transitivity into the homogeneous factorization G_v = G_uv G_vw (Lemma 2.3, from Giudici–Xia), and then systematically ruling out the maximal subgroups of E7(q) and E8(q) using external classifications: parabolic subgroups via the Dynkin diagram and Magma, maximal-rank subgroups from [20], exotic local subgroups from [10], non-parabolic E7(q) subgroups from [12] and [17], and non-parabolic E8(q) subgroups from [18], [10], and [11]. No equation or fitted parameter is reused as a 'prediction': the possible L_v listed in Table 6 are simply the survivors of the case analysis, not inputs to it. The self-citations to the authors' prior papers [31] and [32] are to published general lemmas (e.g., Lemma 2.10 classifying possible factorizations of quasisimple vertex stabilizers) whose assumptions do not include the present theorem; they are independent prior results, not a self-referential uniqueness chain forcing the conclusion. The main genuine concerns are external and computational: the E7(q) non-parabolic classification [12] is an unpublished preprint, Remark 3.9 shows that even the published maximal-rank table [20] contained an error repaired only by an unarchived Magma computation, and many lemmas depend on Magma computations for which no scripts are provided. These are serious reproducibility and exhaustiveness risks, but they are not circularity under the criteria: they do not make the theorem equivalent to its inputs by construction. Hence the circularity score is low.

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

The central claim is a pure existence/non-existence theorem in finite group theory. It introduces no free parameters and no new entities. Its foundation is a stack of external classifications and computational checks: maximal subgroups, factorisations of almost simple groups, maximal-rank subgroups, and Magma verifications. The most fragile item is the unpublished Craven classification [12] for E7(q).

assumptions (9)
  • standard math Classification of the finite simple groups (CFSG)
    Invoked globally; the exceptional-group analysis and the external classification results used in [31, Lemma 3.3] presume the completed classification.
  • domain assumption Liebeck–Seitz classification of maximal subgroups of exceptional groups [18]
    Used in §3 to enumerate all possible core-free maximal vertex stabilizers; a missed class would break the case analysis.
  • domain assumption Cohen–Liebeck–Saxl–Seitz classification of local maximal subgroups [10]
    Basis for the exotic r-local subgroups in case (b) of §3 and for the analysis in Lemma 3.27.
  • domain assumption Craven preprint classification of maximal subgroups of E7(q) [12]
    Section 3.3.1 relies on this not-yet-published preprint for the complete list of non-parabolic, non-maximal-rank E7(q) maximal subgroups.
  • domain assumption Liebeck–Praeger–Saxl classification of maximal factorisations of finite simple groups [19]
    Used throughout §3.2–3.3 to assert the existence or non-existence of factorisations with prescribed factor structures.
  • domain assumption Liebeck–Saxl–Seitz classification of subgroups of maximal rank [20]
    Tables 3 and 5, including module irreducibility facts used in Lemmas 3.7, 3.15–3.20, come from this classification.
  • standard math Zsigmondy's theorem (Lemma 2.2)
    Used repeatedly to choose primitive prime divisors r with r > nf so that |Out(L)|_r = 1.
  • domain assumption Yin–Feng–Xia classification of possible quasisimple vertex stabilizers [31, Lemma 3.3]
    Used in Lemmas 3.5, 3.10, 3.22, and 3.27 to exclude several stabilizer types and to constrain possible (G_v, G_uv, G_vw) factorisations.
  • domain assumption Correctness of Magma computations [2]
    Dozens of finite group checks are asserted on the basis of Magma output, but no scripts or logs are shipped, so the proof depends on those computations being correct and faithfully reported.

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Pith. "Pith review of Exceptional groups and the s-arc-transitivity of vertex-primitive digraphs, II." pith.science (2026). https://pith.science/paper/PHCNGFAS

@misc{pith2026260714603,
  author       = {Pith},
  title        = {Pith review of: Exceptional groups and the s-arc-transitivity of vertex-primitive digraphs, II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHCNGFAS}},
  note         = {Machine review of arXiv:2607.14603}
}
abstract

In this paper, we study the primitive actions of almost simple groups with socle \(E_7(q)\) or \(E_8(q)\) on an \(s\)-arc-transitive digraph. Our motivation goes back to the question of whether \(s\) is bounded above for finite connected \(G\)-vertex-primitive \(s\)-arc-transitive digraphs that are not directed cycles. The question has been reduced by Giudici and Xia to the case where \(G\) is almost simple. This work succeeds our 2025 paper (Yin and Chen), which addressed \({}^3\!D_4(q)\), \(G_2(q)\), \({}^2\!F_4(q)'\), \(F_4(q)\), \(E_6(q)\), and \({}^2\!E_6(q)\). Together with Chen, Giudici, and Praeger's work on \({}^2\!B_2(q)\) and \({}^2\!G_2(q)\), it answers the question for all exceptional groups.

Figures

Figures reproduced from arXiv: 2607.14603 by the authors.

Figure 1
Figure 1. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 1
Figure 1. Dynkin diagrams of Eℓ (ℓ = 7, 8) α1 α3 α4 α5 α2 αℓ Eℓ (ℓ = 7, 8) In Subsection 3.1, we consider parabolic subgroups. The remaining two subsections deal with non-parabolic subgroups of E7(q) and E8(q), respectively. 3.1. Parabolic maximal subgroups. In this subsection, we show that, under Hypothesis 3.1, the case where Gv is a maximal parabolic subgroup cannot occur. A similar result was established for L ∈ {3D4(q), … view at source ↗

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