Rigid automorphisms of linking systems of finite groups of Lie type

Jonathon Villareal

arxiv: 2509.21139 · v2 · submitted 2025-09-25 · 🧮 math.GR

Rigid automorphisms of linking systems of finite groups of Lie type

Jonathon Villareal This is my paper

Pith reviewed 2026-05-18 14:21 UTC · model grok-4.3

classification 🧮 math.GR

keywords rigid automorphismslinking systemsfusion systemsgroups of Lie typequasisimple groupsprime 2orthogonal groupsprojective special linear groups

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The pith

Finite groups of Lie type in odd characteristic have linking systems at p=2 whose rigid automorphisms are all inner except for two families.

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

The paper determines the rigid automorphisms for centric linking systems of finite groups of Lie type in odd characteristic at the prime 2. After reducing via prior results on other quasisimple cases, it finds that these systems have no noninner rigid automorphisms except for the two-dimensional projective special linear groups and even-dimensional orthogonal groups with quadratic forms of nonsquare discriminant. A sympathetic reader cares because a rigid automorphism of the linking system corresponds to an automorphism of the underlying group that centralizes the Sylow 2-subgroup, clarifying how group automorphisms interact with the fusion data. If correct, the result means that for nearly all such groups any automorphism of the linking system fixing the fusion system arises only from conjugation by a central element in the Sylow subgroup.

Core claim

Let L be a centric linking system associated to a saturated fusion system on a finite p-group S. An automorphism of L is rigid if it restricts to the identity on the fusion system. Inner rigid automorphisms are those given by conjugation by an element of the center of S. For the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic, all rigid automorphisms are inner except in the cases of the two-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant.

What carries the argument

Rigid automorphism of a centric linking system, defined as an automorphism that restricts to the identity on the associated fusion system.

If this is right

For all but the two exceptional families, every rigid automorphism of the linking system at p=2 is inner.
Any automorphism of the underlying group G that centralizes the Sylow 2-subgroup S must be inner in the non-exceptional cases.
The classification of noninner rigid automorphisms is now complete for all known quasisimple linking systems at the prime 2.
The result extends the known fact that all rigid automorphisms are inner for odd primes to most of the remaining p=2 cases.

Where Pith is reading between the lines

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

The same reduction strategy might apply to classify rigid automorphisms for linking systems of nonsimple groups or at other small primes.
Small-order computational checks on specific groups like PSL(3,5) or certain orthogonal groups could independently verify the claimed absence of noninner examples.
The two exceptional families may share geometric features, such as low rank or special properties of the quadratic form, that allow extra centralizing automorphisms.
These findings could constrain the possible outer automorphism groups of the associated saturated fusion systems.

Load-bearing premise

The reduction to the Lie-type-in-odd-characteristic case is valid and all prior results on other families of quasisimple linking systems at p=2 apply without further exceptions.

What would settle it

An explicit construction or computation of a noninner rigid automorphism for the linking system of a group of Lie type in odd characteristic that lies outside the two listed families.

read the original abstract

Let $\mathcal{L}$ be a centric linking system associated to a saturated fusion system on a finite $p$-group $S$. An automorphism of $\mathcal{L}$ is said to be rigid if it restricts to the identity on the fusion system. An inner rigid automorphism is conjugation by some element of the center of $S$. If $\mathcal{L}$ is the centric linking system of a finite group $G$, then rigid automorphisms of $\mathcal{L}$ are closely related to automorphisms of $G$ that centralize $S$. For odd primes, all rigid automorphisms are known to be inner, but this fails for the prime 2. We determine which known quasisimple linking systems at the prime 2 have a noninner rigid automorphism. Based on previous results, this reduces to handling the case of the linking systems at the prime 2 of finite simple groups of Lie type in odd characteristic. These have no noninner rigid automorphisms with two families of exceptions: the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant.

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.

Desk Editor's Note private letter to a colleague

This paper closes the p=2 classification of rigid automorphisms for quasisimple linking systems by handling the Lie type case with two exception families.

read the letter

This paper closes the classification for rigid automorphisms of quasisimple linking systems at p=2. It reduces to the Lie type groups in odd characteristic and shows no noninner ones except for the two-dimensional projective special linear groups and even-dimensional orthogonal groups with nonsquare discriminant. What it does well is the focused handling of this last case through direct verification on the known families. The exceptions are pinned down explicitly, which makes the result usable for anyone following up on outer automorphisms or fusion systems. The soft spot is the reliance on prior results for the reduction. If those earlier classifications missed something that interacts with the centric linking systems here, there might be more exceptions, but the paper appears to apply them carefully. The case analysis for the exceptions needs to be solid, but since it's finite groups of Lie type, it should be manageable. This is for people in the fusion systems community or those studying p-local structure at the prime 2. A reader who knows the previous papers on this topic will find it a natural next step. It deserves a serious referee to verify the details of the reduction and the exception checks. I recommend putting it through peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript classifies rigid automorphisms of centric linking systems associated to saturated fusion systems on finite 2-groups arising from quasisimple groups. Building on prior results for other families, it reduces to the case of finite simple groups of Lie type in odd characteristic and concludes that noninner rigid automorphisms exist only for the 2-dimensional projective special linear groups and even-dimensional orthogonal groups for quadratic forms of nonsquare discriminant.

Significance. If the central claim holds, the work completes the determination of which known quasisimple linking systems at p=2 admit noninner rigid automorphisms. This provides a uniform picture of when rigid automorphisms of linking systems are inner, with direct implications for the outer automorphism groups of the underlying fusion systems and for automorphisms of the corresponding finite groups that centralize a Sylow 2-subgroup. The explicit reduction to the odd-characteristic Lie-type case and the identification of two concrete exception families constitute the main contribution.

major comments (1)

[Abstract (reduction paragraph)] Abstract, reduction paragraph: The argument that the classification reduces to the Lie-type-in-odd-characteristic case at p=2 invokes prior results on all other families of quasisimple linking systems without new exceptions. However, the definition of a rigid automorphism (restricting to the identity on the underlying fusion system) may interact with the centric linking system construction for these groups in ways that produce additional noninner examples if the prior results contain implicit assumptions about the 2-group S or its center that do not transfer directly. This reduction step is load-bearing for the central claim and requires explicit verification that no new exceptions arise from the specific form of the linking system in the odd-characteristic setting.

minor comments (1)

[Abstract] The abstract states the main result cleanly but does not indicate the length or structure of the case analysis that follows the reduction; a brief sentence outlining the organization of the proof would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on the manuscript. We address the major comment below and will strengthen the exposition of the reduction argument in the revised version.

read point-by-point responses

Referee: Abstract, reduction paragraph: The argument that the classification reduces to the Lie-type-in-odd-characteristic case at p=2 invokes prior results on all other families of quasisimple linking systems without new exceptions. However, the definition of a rigid automorphism (restricting to the identity on the underlying fusion system) may interact with the centric linking system construction for these groups in ways that produce additional noninner examples if the prior results contain implicit assumptions about the 2-group S or its center that do not transfer directly. This reduction step is load-bearing for the central claim and requires explicit verification that no new exceptions arise from the specific form of the linking system in the odd-characteristic setting.

Authors: We thank the referee for this observation. The reduction rests on the fact that the cited prior works already classify rigid automorphisms for the centric linking systems of all other families of quasisimple groups at p=2, using precisely the same definition (an automorphism of the linking system that restricts to the identity on the underlying saturated fusion system). Those papers treat the standard centric linking system associated to the fusion system arising from the group, and the structural hypotheses on the Sylow 2-subgroup S and its center Z(S) that appear in the statements are satisfied by the groups in question. Because the centric linking system is functorially determined by the fusion system, and the prior results apply to exactly these linking systems, no additional noninner rigid automorphisms are introduced by the construction. We will add a short clarifying paragraph (or subsection) in the introduction that explicitly recalls the relevant hypotheses from the cited works and confirms they hold in the present setting, thereby making the reduction fully self-contained. revision: yes

Circularity Check

0 steps flagged

Relies on prior published results for other quasisimple families at p=2

full rationale

The derivation reduces the classification to the Lie-type-in-odd-characteristic case at p=2 by citing previous results on all other families of quasisimple linking systems. This is a standard external reduction rather than a self-definitional loop or fitted input renamed as prediction inside the paper. No equations or definitions within the current work are shown to be equivalent to the target claim by construction, and the cited priors are treated as independent support. The central claim for the remaining case is handled separately without circular reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard axioms of saturated fusion systems and centric linking systems together with prior classification theorems for other families of groups.

axioms (1)

domain assumption Centric linking systems are associated to saturated fusion systems on finite p-groups
Invoked in the opening definitions of rigid automorphisms.

pith-pipeline@v0.9.0 · 5716 in / 1187 out tokens · 35868 ms · 2026-05-18T14:21:00.583355+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: ... noninner rigid automorphism if and only if ... PSL2(q) ... or PΩ−2n(q) ... ker(μL) cyclic of order 2.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We analyze how automorphisms of G act on the quotient of a Sylow 2-subgroup A ... by the center of F

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Control of fixed points and existence and uniqueness of centric linking systems

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A note on the Schur multiplier of a fusion system

Cambridge Studies in Advanced Mathematics. Cambridge etc.: Cambridge University Press, 1990.isbn: 0-521-37510-X. [Lin06] Markus Linckelmann. “A note on the Schur multiplier of a fusion system”. In: Journal of Algebra296.2 (2006), pp. 402–408. [LO02] Ran Levi and Bob Oliver. “Construction of 2–local finite groups of a type studied by Solomon and Benson”. I...

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Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory

Memoirs of the American Mathematical Society. Providence, RI: American Mathe- matical Society (AMS), 2006.isbn: 978-0-8218-3828-0.doi: 10.1090/memo/0848. [Oli13] Bob Oliver. “Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory”. In:Acta Mathematica211.1 (2013), pp. 141–175. [OV07] Bob Oliver and Joana Ventura. “Extensions o...

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Berlin: Heldermann Verlag, 1992.isbn: 3-88538-009-9

Sigma Series in Pure Mathematics. Berlin: Heldermann Verlag, 1992.isbn: 3-88538-009-9. 38 REFERENCES AppendixA.Tables and diagrams related to the finite groups of Lie type The following tables and diagrams can be found in [GLS05], but are helpful to reference when reading section

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[1] [1]

A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver

[AC10] Michael Aschbacher and Andrew Chermak. “A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver.” In:Ann. Math. (2) 171.2 (2010), pp. 881–978.issn: 0003-486X.doi: 10.4007/annals.2010.171.881. url:annals.princeton.edu/annals/2010/171-2/p06.xhtml. [AKO11] Michael Aschbacher, Radha Kessar, and Bob Oliver.Fusion s...

work page doi:10.4007/annals.2010.171.881 2010

[2] [2]

Fusion systems

London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press, 2011.isbn: 978-1-107-60100-0. [AO16] Michael Aschbacher and Bob Oliver. “Fusion systems”. In:Bulletin of the American Mathematical Society53.4 (2016), pp. 555–615. [AOV12] Kasper KS Andersen, Bob Oliver, and Joana Ventura. “Reduced, tame and exotic fusion systems”. In:...

work page doi:10.1112/plms/pdm057 2011

[3] [3]

Providence, RI: American Mathematical Society (AMS), 2021.isbn: 978-1-4704-5665-8.doi:10.1090/conm/765

Contemporary Mathematics. Providence, RI: American Mathematical Society (AMS), 2021.isbn: 978-1-4704-5665-8.doi:10.1090/conm/765. [Asc86] Michael Aschbacher.Finite group theory. Vol

work page doi:10.1090/conm/765 2021

[4] [4]

Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups

[BLO03] C. Broto, Ran Levi, and B. Oliver. “Homotopy Equivalences of p-Completed Classifying Spaces of Finite Groups”. In:Inventiones Mathematicae151 (Mar. 2003), pp. 611–664.issn: 0020-9910.doi:10.1007/S00222-002-0264-5. [BMO12] Carles Broto, Jesper Møller, and Bob Oliver. “Equivalences between fusion systems of finite groups of Lie type”. In:Journal of ...

work page doi:10.1007/s00222-002-0264-5 2003

[5] [5]

Providence, RI: American Mathematical Society (AMS), 2019.isbn: 978-1-4704- 3772-5.doi:10.1090/memo/1267

Contemporary Mathematics. Providence, RI: American Mathematical Society (AMS), 2019.isbn: 978-1-4704- 3772-5.doi:10.1090/memo/1267. [Bou60] Nicolas Bourbaki.Elements of mathematics. Lie groups and Lie algebras. Chapters 4–6. Transl. from the French by Andrew Pressley. English. Paperback reprint of the hardback edition

work page doi:10.1090/memo/1267 2019

[6] [6]

Subgroup families controlling p-local finite groups

Berlin: Springer, 1960.isbn: 978-3-540-69171-6. [Bro+05] Carles Broto, Natalia Castellana, Jesper Grodal, Ran Levi, and Bob Oliver. “Subgroup families controlling p-local finite groups”. In:Proceedings of the London Mathematical Society91.2 (2005), pp. 325–354. [Bro+23] Carles Broto, Jesper Møller, Bob Oliver, and Albert Ruiz.Realizability and tameness of...

work page 1960

[7] [7]

The Sylow 2-subgroups of the finite classical groups

arXiv: 2102.08278 [math.GR] .url: https: //arxiv.org/abs/2102.08278. 36 REFERENCES [CF64] Roger Carter and Paul Fong. “The Sylow 2-subgroups of the finite classical groups”. In:Journal of Algebra1.2 (1964), pp. 139–151. [Che13] Andrew Chermak. “Fusion systems and localities.” In:Acta Mathematica211.1 (2013), pp. 47–139.issn: 0001-5962.doi:10.1007/s11511-0...

work page doi:10.1007/s11511-013-0099-5 1964

[8] [8]

Control of fixed points and existence and uniqueness of centric linking systems

Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 2011.isbn: 978-1-107-00596-9. [GL16] George Glauberman and Justin Lynd. “Control of fixed points and existence and uniqueness of centric linking systems”. In:Inventiones Mathematicae206.2 (2016), pp. 441–484. [GL21] George Glauberman and Justin Lynd. “Rigid automorphisms of ...

work page 2011

[9] [9]

2021, e23

Cambridge University Press. 2021, e23. [GLS05] Daniel Gorenstein, Richard Lyons, and Ronald Solomon.The classification of the finite simple groups. Vol

work page 2021

[10] [10]

2-fusion in finite groups

[Gol74] David M Goldschmidt. “2-fusion in finite groups”. In:Annals of Mathematics99.1 (1974), pp. 70–117. [Hum90] James E. Humphreys.Reflection groups and Coxeter groups. Vol

work page 1974

[11] [11]

A note on the Schur multiplier of a fusion system

Cambridge Studies in Advanced Mathematics. Cambridge etc.: Cambridge University Press, 1990.isbn: 0-521-37510-X. [Lin06] Markus Linckelmann. “A note on the Schur multiplier of a fusion system”. In: Journal of Algebra296.2 (2006), pp. 402–408. [LO02] Ran Levi and Bob Oliver. “Construction of 2–local finite groups of a type studied by Solomon and Benson”. I...

work page 1990

[12] [12]

Equivalences of classifying spaces completed at odd primes

Cambridge University Press. 1996, pp. 119–137. [Oli04] Bob Oliver. “Equivalences of classifying spaces completed at odd primes”. In: Mathematical Proceedings of the Cambridge Philosophical Society. Vol

work page 1996

[13] [13]

2004, pp

Cambridge University Press. 2004, pp. 321–347. [Oli06] Bob Oliver.Equivalences of classifying spaces completed at the prime two. Vol

work page 2004

[14] [14]

Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory

Memoirs of the American Mathematical Society. Providence, RI: American Mathe- matical Society (AMS), 2006.isbn: 978-0-8218-3828-0.doi: 10.1090/memo/0848. [Oli13] Bob Oliver. “Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory”. In:Acta Mathematica211.1 (2013), pp. 141–175. [OV07] Bob Oliver and Joana Ventura. “Extensions o...

work page doi:10.1090/memo/0848 2006

[15] [15]

Automorphisms of finite linear groups

Prog. Math. Basel: Birkh¨ auser, 2009.isbn: 978-3-7643-9997-9.doi:10.1007/978-3-7643-9998-6. [Ste60] Robert Steinberg. “Automorphisms of finite linear groups”. In:Canadian Journal of Mathematics12 (1960), pp. 606–615. [Suz82] Michio Suzuki.Group theory I. Berlin; New York: Springer-Verlag, 1982.url: http://www.worldcat.org/search?qt=worldcat_org_all&q=354...

work page doi:10.1007/978-3-7643-9998-6 2009

[16] [16]

Berlin: Heldermann Verlag, 1992.isbn: 3-88538-009-9

Sigma Series in Pure Mathematics. Berlin: Heldermann Verlag, 1992.isbn: 3-88538-009-9. 38 REFERENCES AppendixA.Tables and diagrams related to the finite groups of Lie type The following tables and diagrams can be found in [GLS05], but are helpful to reference when reading section

work page 1992