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arxiv: 2304.08459 · v4 · pith:TMAS5E5Hnew · submitted 2023-04-17 · 🧮 math.GR

On transitive sets of derangements in primitive groups

Peter M\"uller This is my paper

Pith reviewed 2026-05-24 09:10 UTC · model grok-4.3

classification 🧮 math.GR
keywords primitive permutation groupsderangementsSteinberg triality group^3D_4(2)Kourovka Notebookfinite simple groupspermutation representations
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The pith

The group ^3D_4(2) has a primitive action of degree 4064256 in which two points are not joined by any derangement.

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

The paper constructs an explicit primitive permutation action of the Steinberg triality group ^3D_4(2) on 4064256 points. Within this action it identifies a pair of distinct points α and β for which no element of the group that fixes no points sends α to β. The construction supplies a concrete negative answer to Thompson's question 8.75 from the Kourovka Notebook. A reader would care because the result shows that the set of derangements in a primitive group need not connect every ordered pair of distinct points.

Core claim

We construct a primitive permutation action of the Steinberg triality group ^3D_4(2) of degree 4064256 and show that there are distinct points α,β such that there is no derangement g∈^3D_4(2) with α^g=β. This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.

What carries the argument

The explicit construction of the faithful primitive action of ^3D_4(2) of degree 4064256 together with the exhaustive computational verification that no derangement maps one chosen point to the other.

If this is right

  • Thompson's question 8.75 in the Kourovka Notebook has a negative answer.
  • The derangements in a primitive permutation group need not act transitively on ordered pairs of distinct points.
  • The phenomenon occurs for this particular action of ^3D_4(2).

Where Pith is reading between the lines

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

  • Similar explicit constructions for other exceptional groups of Lie type could produce further counterexamples.
  • The result indicates that computational enumeration on groups of this size can resolve open questions once an explicit action is available.

Load-bearing premise

The given construction produces a faithful primitive action of the stated degree and the check that no derangement sends α to β is complete and correct.

What would settle it

An explicit derangement in the group that maps the first chosen point to the second chosen point in this action would falsify the claim.

read the original abstract

We construct a primitive permutation action of the Steinberg triality group $^3D_4(2)$ of degree $4064256$ and show that there are distinct points $\alpha,\beta$ such that there is no derangement $g\in{^3D_4}(2)$ with $\alpha^g=\beta$. This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.

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

1 major / 0 minor

Summary. The paper constructs a primitive permutation action of the Steinberg triality group ^3D_4(2) of degree 4064256 and exhibits distinct points α, β such that no derangement in the group maps α to β, thereby giving a negative answer to Thompson's question 8.75 in the Kourovka Notebook.

Significance. If the construction and verification hold, the result resolves an open question on the existence of transitive derangement sets in primitive groups and has implications for the derangement graph and related problems in finite permutation group theory.

major comments (1)
  1. [Abstract and introductory construction paragraph] The manuscript provides no description of the explicit construction of the degree-4064256 action (e.g., the maximal subgroup whose cosets realize the action), the primitivity test employed, or the algorithm used to certify that the coset representatives or orbit-stabilizer computation yields no derangement sending α to β. Given |G| is too large for exhaustive enumeration, this verification method is load-bearing for the central claim and must be specified for reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for greater detail on the construction and verification. We address the major comment below and will revise the manuscript to incorporate the requested information.

read point-by-point responses

  1. Referee: [Abstract and introductory construction paragraph] The manuscript provides no description of the explicit construction of the degree-4064256 action (e.g., the maximal subgroup whose cosets realize the action), the primitivity test employed, or the algorithm used to certify that the coset representatives or orbit-stabilizer computation yields no derangement sending α to β. Given |G| is too large for exhaustive enumeration, this verification method is load-bearing for the central claim and must be specified for reproducibility.

    Authors: We agree that the manuscript as submitted lacked an explicit description of the construction, the primitivity test, and the verification algorithm. In the revised version we will add a dedicated subsection (likely in Section 2) that specifies: (i) the maximal subgroup M < ^3D_4(2) realizing the action on cosets (including its structure and the source of the embedding, e.g., via the Atlas or computational realization); (ii) the primitivity test, which consists of confirming maximality of M by direct computation of its normalizer and known subgroup lattice data; and (iii) the algorithm used to certify that no derangement sends α to β. The latter proceeds by computing the relevant double coset or by using the orbit-stabilizer theorem together with fixed-point-ratio bounds derived from the character table of ^3D_4(2), avoiding exhaustive enumeration of G. These additions will make the central claim fully reproducible. revision: yes

Circularity Check

0 steps flagged

Explicit construction with no self-referential or fitted derivation

full rationale

The paper's central result is an explicit construction of a primitive action of ^3D_4(2) on 4064256 points together with a verification that no derangement sends α to β. The abstract states this directly as a construction answering Thompson's question in the negative. No equations, parameters, or claims are shown to reduce to their own inputs by definition, no fitted quantities are relabeled as predictions, and no load-bearing premise rests on a self-citation chain. The derivation is therefore self-contained as a direct existence proof via construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim depends on standard facts about finite groups of Lie type and their permutation representations; no free parameters or invented entities are introduced.

axioms (2)
  • standard math The Steinberg triality group ^3D_4(2) exists as a finite simple group of Lie type with known order and representation theory.
    Invoked to define the group whose action is constructed.
  • standard math Primitive permutation actions and derangements are well-defined notions in the theory of finite permutation groups.
    Background definitions used throughout the claim.

pith-pipeline@v0.9.0 · 5585 in / 1297 out tokens · 28716 ms · 2026-05-24T09:10:02.846044+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    Atlas of finite group representations - Version 3 , https://brauer.maths.qmul.ac.uk/Atlas/v3/

    work page

  2. [2]

    Bosma, J

    W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), no. 3-4, 235–265

    work page 1997

  3. [3]

    T. C. Burness and M. Giudici, Classical groups, derangements and primes, Australian Mathematical Society Lecture Series, vol. 25, Cam- bridge University Press, Cambridge, 2016

    work page 2016

  4. [4]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A . Wilson, Atlas of finite groups , Oxford University Press, Eynsham, 1985, Maximal subgroups and ordinary characters for simple groups, Wit h computational assistance from J. G. Thackray

    work page 1985

  5. [5]

    J. D. Dixon and B. Mortimer, Permutation groups , Springer–Verlag, New York, 1996

    work page 1996

  6. [6]

    The GAP Group, GAP – Groups, Algorithms, and Programming, Ver- sion 4.12.2 , 2022. 4

    work page 2022

  7. [7]

    R. M. Guralnick and J. Saxl, Primitive permutation characters , Groups, combinatorics & geometry (Durham, 1990), London Math. Soc. Le cture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 3 64– 367

    work page 1990

  8. [8]

    D. G. Higman, Intersection matrices for finite permutation groups , J. Algebra 6 (1967), 22–42

    work page 1967

  9. [9]

    Kaplan and A

    G. Kaplan and A. Lev, Transversals in permutation groups and factori- sations of complete graphs , Bull. Austral. Math. Soc. 65 (2002), no. 2, 277–288

    work page 2002

  10. [10]

    E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka note- book, Sobolev Institute of Mathematics. Russian Academy of Sciences. Siberian Branch, Novosibirsk, 2018, Unsolved problems in group the ory, Nineteenth edition [ MR0204500], March 2019 update

    work page 2018

  11. [11]

    Kleidman, The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups , J

    P. Kleidman, The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups , J. Algebra 115 (1988), 182– 199

    work page 1988

  12. [12]

    The Sage Developers, SageMath, the Sage Mathematics Software Sys- tem, 2022, DOI 10.5281/zenodo.6259615. 5

    work page doi:10.5281/zenodo.6259615 2022