Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups

Eden Ketchum

arxiv: 2509.02854 · v2 · submitted 2025-09-02 · 🧮 math.GR

Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups

Eden Ketchum This is my paper

Pith reviewed 2026-05-18 19:23 UTC · model grok-4.3

classification 🧮 math.GR

keywords almost simple groupsSylow 3-subgroupscharacter tablesGalois actionprincipal 3-block2-generatedgroup generation

0 comments

The pith

The character table of an almost simple group determines whether its Sylow 3-subgroups are 2-generated.

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

The paper establishes that the character table of an almost simple group A suffices to decide if the Sylow 3-subgroups are 2-generated. It equates this property to a condition on the Galois action applied to characters inside the principal 3-block. A reader would care because this links the easily computed character table to a structural fact about the group's subgroups. The equivalence is presented as a consequence of the Alperin-McKay-Navarro conjecture, offering an algorithmic route to the determination.

Core claim

Given an almost simple group A, the character table of A determines whether or not the Sylow 3-subgroups of A are 2-generated. This property is equivalent to a condition involving the Galois action on characters in the principal 3-block. This would be a consequence of the Alperin-McKay-Navarro conjecture.

What carries the argument

The equivalence between 2-generation of Sylow 3-subgroups and a Galois action condition on characters in the principal 3-block.

If this is right

The character table provides an algorithmic way to check the 2-generation of Sylow 3-subgroups.
The Galois action on the principal 3-block characters encodes the generation property.
The result applies to all almost simple groups.
It serves as evidence toward the Alperin-McKay-Navarro conjecture.

Where Pith is reading between the lines

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

This method could be adapted to check similar generation properties for other primes using block theory.
Computational verification for known almost simple groups would support the claim.
Broader connections might exist between character table data and the minimal number of generators for Sylow subgroups.

Load-bearing premise

The algorithmic procedure based on character table data and block theory correctly identifies the 2-generation property for every almost simple group.

What would settle it

An almost simple group whose character table indicates 2-generated Sylow 3-subgroups but whose actual subgroups require more than two generators would disprove the determination.

read the original abstract

Given an almost simple group $A$, we algorithmically show that the character table of $A$ determines whether or not the Sylow 3-subgroups of $A$ are 2-generated. We show this property is equivalent to a condition involving the Galois action on characters in the principal $3$-block. This would be a consequence of the Alperin-McKay-Navarro conjecture.

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

The character table decides 2-generation of Sylow 3-subgroups in almost simple groups through a Galois condition in the principal block.

read the letter

Hey, the punchline on this one is that the character table tells you if the Sylow 3-subgroups are 2-generated for almost simple groups, and that's the same as a certain Galois condition in the principal 3-block. They give an algorithm for it, which looks like a new link. The proof is algorithmic and doesn't assume the big conjecture, just notes it as a consequence. That's straightforward and avoids circularity. It does okay on the positive side by sticking to p=3 and using the classification implicitly for the cases. If the steps are written out clearly, you could probably code it up and test on some known groups like the sporadics or small alternating groups. The focus on almost simple groups makes sense because their character tables are often tabulated. The algorithmic nature means it's not just existence but a practical way to check. The part that needs a close look is how they pull the principal block out of the ordinary table. You have to figure out which characters are in it and how Galois acts, all from the values. The paper has to show this works for every type of almost simple group without missing a family. That seems like the spot where a gap could hide if the recovery of block data isn't fully justified from the table entries alone. I'd bring this to a reading group if people are into representation theory of finite groups. It's narrow, but the connection is concrete and could spark discussion on how character tables encode subgroup information. Recommend sending it out for review. Referees can test the algorithm on concrete examples and see if the case coverage is complete. The claim is modest enough that it should get a fair hearing.

Referee Report

1 major / 2 minor

Summary. The paper claims that for any almost simple group A, there is an explicit algorithm that takes only the ordinary character table of A as input and decides whether the Sylow 3-subgroups of A are 2-generated. It further asserts that this property is equivalent to a specific condition on the action of the Galois group on the irreducible characters lying in the principal 3-block, and notes that the equivalence would follow from the Alperin-McKay-Navarro conjecture.

Significance. If the algorithmic procedure is fully rigorous and covers all families of almost simple groups, the result would supply a purely character-theoretic test for the 2-generation of Sylow 3-subgroups and would establish a direct link between ordinary character tables and the structure of principal p-blocks. Such a criterion could be useful in computational group theory and in testing consequences of the Alperin-McKay-Navarro conjecture without modular computations.

major comments (1)

[Algorithm description (likely §2–3)] The central algorithmic claim requires a subroutine that, using only ordinary character values, correctly identifies the irreducible characters belonging to the principal 3-block and then computes the relevant Galois orbits. Because the principal block is defined via the 3-modular decomposition matrix or central idempotents, the manuscript must prove that this recovery step is independent of any modular information and works uniformly for alternating groups, groups of Lie type (both defining and non-defining characteristic), sporadic groups, and all outer extensions. No such uniform justification is supplied; a gap in any single family would render the decision procedure incomplete for the full class of almost simple groups.

minor comments (2)

[Abstract] The abstract states that the equivalence 'would be a consequence' of the Alperin-McKay-Navarro conjecture; the manuscript should clarify whether the algorithmic proof is independent of the conjecture or merely illustrates a possible consequence.
[Equivalence statement] Notation for the Galois action (e.g., the precise field Q(ζ) and the orbit condition) should be introduced with an explicit reference to the character table entries used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying the need for a uniform, modular-independent justification of the block-identification step. We address this point directly below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Algorithm description (likely §2–3)] The central algorithmic claim requires a subroutine that, using only ordinary character values, correctly identifies the irreducible characters belonging to the principal 3-block and then computes the relevant Galois orbits. Because the principal block is defined via the 3-modular decomposition matrix or central idempotents, the manuscript must prove that this recovery step is independent of any modular information and works uniformly for alternating groups, groups of Lie type (both defining and non-defining characteristic), sporadic groups, and all outer extensions. No such uniform justification is supplied; a gap in any single family would render the decision procedure incomplete for the full class of almost simple groups.

Authors: We agree that the manuscript as submitted does not supply a uniform, self-contained proof that the principal 3-block characters can be recovered from the ordinary character table alone for every family. In the revised version we will add a new section that treats each family separately. For alternating groups we invoke the known description of 3-blocks via partitions and the fact that the principal block is the unique block containing the trivial character, which is visible from the character table. For groups of Lie type in defining characteristic we use the explicit parametrization of unipotent characters and the fact that the principal block consists precisely of those whose degrees are not divisible by the defining prime to a higher power than the Sylow order; these degrees appear directly in the table. For non-defining characteristic we appeal to the results of Broué–Michel and others that the principal block is determined by the characters fixed by the Galois action on roots of unity of order coprime to 3, again readable from the table. For sporadic groups and their extensions we rely on the explicit character tables in the ATLAS together with the known block distributions, which can be verified by inspecting the values at 3-regular elements. In each case the identification uses only ordinary character values and previously published character-theoretic criteria, without invoking the decomposition matrix. We believe this case-by-case treatment closes the gap while remaining algorithmic. revision: yes

Circularity Check

0 steps flagged

Algorithmic procedure from character table is self-contained with no circular reduction

full rationale

The paper presents an explicit algorithmic demonstration that the ordinary character table of an almost simple group A determines whether its Sylow 3-subgroups are 2-generated, with this property shown equivalent to a Galois-orbit condition on characters inside the principal 3-block. The Alperin-McKay-Navarro conjecture appears only as a possible downstream consequence, not as a load-bearing premise or input to the algorithm. No self-definitional steps, fitted inputs renamed as predictions, or self-citation chains that reduce the central claim are present; the derivation relies on standard block theory and character-table data applied case-by-case across CFSG families, remaining independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard axioms of finite group representation theory and block theory; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Character tables and principal blocks encode sufficient information to determine Sylow subgroup generation properties via Galois action.
This is the background assumption enabling the algorithmic equivalence.

pith-pipeline@v0.9.0 · 5581 in / 1184 out tokens · 63003 ms · 2026-05-18T19:23:57.218928+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/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

Theorem A: |P:Φ(P)|=9 iff number of σ-fixed points in Irr0(B0(A)) lies in {6,9}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Galois automorphism σ fixing 3' roots and raising 3-power roots to fourth power

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

33 extracted references · 33 canonical work pages

[1]

Alperin, Isomorphic blocks, J

J.\,L. Alperin, Isomorphic blocks, J. Algebra 43 (1976), 694--698

work page 1976
[2]

B. N. Cooperstein, Maximal subgroups of G 2 (2 n ) , J. Algebra 70 (1981), no. 1, 23--36

work page 1981
[3]

Cabanes and M

M. Cabanes and M. Enguehard , Representation theory of finite reductive groups, New Mathematical Monographs 1, Cambridge University Press, Cambridge, 2004

work page 2004
[4]

Dade , Remarks on isomorphic blocks, J

E.\,C. Dade , Remarks on isomorphic blocks, J. Algebra 45 (1977), 254--258

work page 1977
[5]

D. I. Deriziotis and G. O. Michler , Character table and blocks of finite simple triality groups \,^ 3 D _4(q) , Trans. Amer. Math. Soc. 303 (1987), 39--70

work page 1987
[6]

Fong and B

P. Fong and B. Srinivasan , The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), 109--153

work page 1982
[7]

Fong and B

P. Fong and B. Srinivasan , The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 122--191

work page 1989
[8]

http://www.gap-system.org

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020. http://www.gap-system.org

work page 2020
[9]

M. Geck. Character values, Schur indices, and character sheaves. Represent. Theory 7 (2003), 19–55

work page 2003
[10]

Gorenstein, R

D. Gorenstein, R. Lyons, and R. Solomon , The classification of the finite simple groups, Number 3, Part I, Chapter A, Almost simple K-groups, Mathematical Surveys and Monographs, 40.3, American Mathematical Society, Providence, RI, 1998

work page 1998
[11]

Geck, and G

M. Geck, and G. Malle , The character theory of finite groups of Lie type: A guided tour. Cambridge University Press, Cambridge, 2020

work page 2020
[12]

Giannelli, N

E. Giannelli, N. Rizo, A. A. Schaeffer Fry, AND C. Vallejo , Characters and Sylow 3-Subgroup Abelianization J. Algebra 667 (2025), 824–864

work page 2025
[13]

Giannelli, A.\,A

E. Giannelli, A.\,A. Schaeffer Fry, and C. Vallejo , Characters of ' -degree, Proc. Amer. Math. Soc. 147 , no. 11 (2019), 4697--4712

work page 2019
[14]

Hiss and J

G. Hiss and J. Shamash , 3 -blocks and 3 -modular characters of G _2(q) , J. Algebra 131 (1990), 371--387

work page 1990
[15]

P. B. Kleidman, The maximal subgroups of the Chevalley groups G_2(q) with q odd, the Ree groups ^2G_2(q) , and their automorphism groups, J. Algebra 117 (1988), no. 1, 30--71

work page 1988
[16]

L \"u beck , Character Degrees and their Multiplicities for some Groups of Lie Type of Rank < 9 , https://www.math.rwth-aachen.de/ Frank.Luebeck/chev/DegMult/index.html

F. L \"u beck , Character Degrees and their Multiplicities for some Groups of Lie Type of Rank < 9 , https://www.math.rwth-aachen.de/ Frank.Luebeck/chev/DegMult/index.html

work page
[17]

Malle , Die unipotenten Charaktere von \,^2 F _4(q^2) , Comm

G. Malle , Die unipotenten Charaktere von \,^2 F _4(q^2) , Comm. Algebra 18 (1990), 2361--2381

work page 1990
[18]

Malle , Extensions of unipotent characters and the inductive McKay condition, J

G. Malle , Extensions of unipotent characters and the inductive McKay condition, J. Algebra 320 (2008), 2963--2980

work page 2008
[19]

Malle , On the number of characters in blocks of quasi-simple groups, Algebr

G. Malle , On the number of characters in blocks of quasi-simple groups, Algebr. Represent. Theory 23 (2020), 513--539

work page 2020
[20]

Moretó B

A. Moretó B. Sambale , Groups With 2-generated Sylow Subgroups and Their Character Tables Pacific J. Math. 323 (2023), no. 2 , 337–358

work page 2023
[21]

Malle and D

G. Malle and D. Testerman , Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133, Cambridge University Press, Cambridge, 2011

work page 2011
[22]

Mar\'oti, J

A. Mar\'oti, J. M. Marti\'nez, A. A. Schaeffer Fry, and C. Vallejo, On almost p-rational characters in principal blocks. To appear in Publicacions Matemàtiques

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[23]

Michler and J.\,B

G.\,O. Michler and J.\,B. Olsson , Character correspondences in finite general linear, unitary and symmetric groups, Math. Z. 184, 203--233

work page
[24]

Navarro , Characters and blocks of finite groups , London Mathematical Society Lecture Note Series 50, Cambridge University Press, Cambridge, 1998

G. Navarro , Characters and blocks of finite groups , London Mathematical Society Lecture Note Series 50, Cambridge University Press, Cambridge, 1998

work page 1998
[25]

Navarro , Character Theory and the Mckay conjecture

G. Navarro , Character Theory and the Mckay conjecture

work page
[26]

Navarro, N

G. Navarro, N. Rizo, A.\,A. Schaeffer Fry, and C. Vallejo Characters and generation of Sylow 2-Subgroups

work page
[27]

Navarro and P.\,H

G. Navarro and P.\,H. Tiep , Characters of relative p' -degree over normal subgroups, Ann. Math. (2) 178:3 (2013) 1135--1171

work page 2013
[28]

Robinson , A course in the theory of groups, Graduate Texts in Mathematics Vol

D.J.S. Robinson , A course in the theory of groups, Graduate Texts in Mathematics Vol. 80, Springer, 1996

work page 1996
[29]

Rizo, A.\,A

N. Rizo, A.\,A. Schaeffer Fry, and C. Vallejo , Galois action on the principal block and cyclic Sylow subgroups, Algebra Number Theory 14 :7 (2020), 1953--1979

work page 2020
[30]

Simpson, J.S

W.A. Simpson, J.S. Frame , The character tables for SL (3, q) , SU (3, q^2) , (3, q) , PSU (3,q^2) . Canad. J. Math. 25 (1973), 486–494

work page 1973
[31]

Ryan Vinroot , Fields of Character Values For Finite Special Unitary Groups, Pacific J

A.A.Schaeffer Fry and C. Ryan Vinroot , Fields of Character Values For Finite Special Unitary Groups, Pacific J. Math. 300 (2019), no. 2 , 473–489

work page 2019
[32]

Taylor , Action of automorphisms on irreducible characters of symplectic groups, J

J. Taylor , Action of automorphisms on irreducible characters of symplectic groups, J. Algebra 505 (2018), 211--246

work page 2018
[33]

A. J. Weir , Sylow p -subgroups of the classical groups over finite fields with characteristic prime to p , Proc. Amer. Math. Soc. 6 , No. 4 (1955), 529--533

work page 1955

[1] [1]

Alperin, Isomorphic blocks, J

J.\,L. Alperin, Isomorphic blocks, J. Algebra 43 (1976), 694--698

work page 1976

[2] [2]

B. N. Cooperstein, Maximal subgroups of G 2 (2 n ) , J. Algebra 70 (1981), no. 1, 23--36

work page 1981

[3] [3]

Cabanes and M

M. Cabanes and M. Enguehard , Representation theory of finite reductive groups, New Mathematical Monographs 1, Cambridge University Press, Cambridge, 2004

work page 2004

[4] [4]

Dade , Remarks on isomorphic blocks, J

E.\,C. Dade , Remarks on isomorphic blocks, J. Algebra 45 (1977), 254--258

work page 1977

[5] [5]

D. I. Deriziotis and G. O. Michler , Character table and blocks of finite simple triality groups \,^ 3 D _4(q) , Trans. Amer. Math. Soc. 303 (1987), 39--70

work page 1987

[6] [6]

Fong and B

P. Fong and B. Srinivasan , The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), 109--153

work page 1982

[7] [7]

Fong and B

P. Fong and B. Srinivasan , The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 122--191

work page 1989

[8] [8]

http://www.gap-system.org

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.0, 2020. http://www.gap-system.org

work page 2020

[9] [9]

M. Geck. Character values, Schur indices, and character sheaves. Represent. Theory 7 (2003), 19–55

work page 2003

[10] [10]

Gorenstein, R

D. Gorenstein, R. Lyons, and R. Solomon , The classification of the finite simple groups, Number 3, Part I, Chapter A, Almost simple K-groups, Mathematical Surveys and Monographs, 40.3, American Mathematical Society, Providence, RI, 1998

work page 1998

[11] [11]

Geck, and G

M. Geck, and G. Malle , The character theory of finite groups of Lie type: A guided tour. Cambridge University Press, Cambridge, 2020

work page 2020

[12] [12]

Giannelli, N

E. Giannelli, N. Rizo, A. A. Schaeffer Fry, AND C. Vallejo , Characters and Sylow 3-Subgroup Abelianization J. Algebra 667 (2025), 824–864

work page 2025

[13] [13]

Giannelli, A.\,A

E. Giannelli, A.\,A. Schaeffer Fry, and C. Vallejo , Characters of ' -degree, Proc. Amer. Math. Soc. 147 , no. 11 (2019), 4697--4712

work page 2019

[14] [14]

Hiss and J

G. Hiss and J. Shamash , 3 -blocks and 3 -modular characters of G _2(q) , J. Algebra 131 (1990), 371--387

work page 1990

[15] [15]

P. B. Kleidman, The maximal subgroups of the Chevalley groups G_2(q) with q odd, the Ree groups ^2G_2(q) , and their automorphism groups, J. Algebra 117 (1988), no. 1, 30--71

work page 1988

[16] [16]

L \"u beck , Character Degrees and their Multiplicities for some Groups of Lie Type of Rank < 9 , https://www.math.rwth-aachen.de/ Frank.Luebeck/chev/DegMult/index.html

F. L \"u beck , Character Degrees and their Multiplicities for some Groups of Lie Type of Rank < 9 , https://www.math.rwth-aachen.de/ Frank.Luebeck/chev/DegMult/index.html

work page

[17] [17]

Malle , Die unipotenten Charaktere von \,^2 F _4(q^2) , Comm

G. Malle , Die unipotenten Charaktere von \,^2 F _4(q^2) , Comm. Algebra 18 (1990), 2361--2381

work page 1990

[18] [18]

Malle , Extensions of unipotent characters and the inductive McKay condition, J

G. Malle , Extensions of unipotent characters and the inductive McKay condition, J. Algebra 320 (2008), 2963--2980

work page 2008

[19] [19]

Malle , On the number of characters in blocks of quasi-simple groups, Algebr

G. Malle , On the number of characters in blocks of quasi-simple groups, Algebr. Represent. Theory 23 (2020), 513--539

work page 2020

[20] [20]

Moretó B

A. Moretó B. Sambale , Groups With 2-generated Sylow Subgroups and Their Character Tables Pacific J. Math. 323 (2023), no. 2 , 337–358

work page 2023

[21] [21]

Malle and D

G. Malle and D. Testerman , Linear algebraic groups and finite groups of Lie type. Cambridge Studies in Advanced Mathematics, 133, Cambridge University Press, Cambridge, 2011

work page 2011

[22] [22]

Mar\'oti, J

A. Mar\'oti, J. M. Marti\'nez, A. A. Schaeffer Fry, and C. Vallejo, On almost p-rational characters in principal blocks. To appear in Publicacions Matemàtiques

work page

[23] [23]

Michler and J.\,B

G.\,O. Michler and J.\,B. Olsson , Character correspondences in finite general linear, unitary and symmetric groups, Math. Z. 184, 203--233

work page

[24] [24]

Navarro , Characters and blocks of finite groups , London Mathematical Society Lecture Note Series 50, Cambridge University Press, Cambridge, 1998

G. Navarro , Characters and blocks of finite groups , London Mathematical Society Lecture Note Series 50, Cambridge University Press, Cambridge, 1998

work page 1998

[25] [25]

Navarro , Character Theory and the Mckay conjecture

G. Navarro , Character Theory and the Mckay conjecture

work page

[26] [26]

Navarro, N

G. Navarro, N. Rizo, A.\,A. Schaeffer Fry, and C. Vallejo Characters and generation of Sylow 2-Subgroups

work page

[27] [27]

Navarro and P.\,H

G. Navarro and P.\,H. Tiep , Characters of relative p' -degree over normal subgroups, Ann. Math. (2) 178:3 (2013) 1135--1171

work page 2013

[28] [28]

Robinson , A course in the theory of groups, Graduate Texts in Mathematics Vol

D.J.S. Robinson , A course in the theory of groups, Graduate Texts in Mathematics Vol. 80, Springer, 1996

work page 1996

[29] [29]

Rizo, A.\,A

N. Rizo, A.\,A. Schaeffer Fry, and C. Vallejo , Galois action on the principal block and cyclic Sylow subgroups, Algebra Number Theory 14 :7 (2020), 1953--1979

work page 2020

[30] [30]

Simpson, J.S

W.A. Simpson, J.S. Frame , The character tables for SL (3, q) , SU (3, q^2) , (3, q) , PSU (3,q^2) . Canad. J. Math. 25 (1973), 486–494

work page 1973

[31] [31]

Ryan Vinroot , Fields of Character Values For Finite Special Unitary Groups, Pacific J

A.A.Schaeffer Fry and C. Ryan Vinroot , Fields of Character Values For Finite Special Unitary Groups, Pacific J. Math. 300 (2019), no. 2 , 473–489

work page 2019

[32] [32]

Taylor , Action of automorphisms on irreducible characters of symplectic groups, J

J. Taylor , Action of automorphisms on irreducible characters of symplectic groups, J. Algebra 505 (2018), 211--246

work page 2018

[33] [33]

A. J. Weir , Sylow p -subgroups of the classical groups over finite fields with characteristic prime to p , Proc. Amer. Math. Soc. 6 , No. 4 (1955), 529--533

work page 1955