Subnormalizers and character correspondences in $p$-solvable groups

Gabriel A. L. Souza

arxiv: 2605.22669 · v1 · pith:SWDYBPE5new · submitted 2026-05-21 · 🧮 math.RT · math.GR

Subnormalizers and character correspondences in p-solvable groups

Gabriel A. L. Souza This is my paper

Pith reviewed 2026-05-22 03:57 UTC · model grok-4.3

classification 🧮 math.RT math.GR

keywords subnormalizer conjecturep-solvable groupscharacter correspondencesGlauberman correspondencelocal-global conjecturesrepresentation theoryfinite groupsp-length

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The pith

The strong subnormalizer conjecture holds for p-solvable groups with odd p when the subnormalizer subset forms a subgroup.

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 strong subnormalizer conjecture is valid in p-solvable groups for odd primes p, provided the subnormalizer subset of an irreducible character is itself a subgroup. It further confirms the conjecture for all such groups that have p-length exactly one. A reader cares because these local-global conjectures aim to link the behavior of subgroups and normalizers to the global distribution of irreducible characters, and proving them in broad families of groups advances the program of reducing representation-theoretic questions to local data.

Core claim

In a p-solvable group with odd prime p, if the subnormalizer subset attached to an irreducible character is a subgroup, then the strong subnormalizer conjecture holds. The same conclusion is obtained without that subgroup hypothesis whenever the p-length equals one. The proofs also produce additional relations satisfied by the Glauberman correspondence in these groups.

What carries the argument

The subnormalizer subset of an irreducible character, required to be a subgroup, which controls the correspondence between characters of the group and those of certain subgroups or quotients via the Glauberman map.

If this is right

The strong subnormalizer conjecture is settled for every p-solvable group of p-length one when p is odd.
New compatibility relations between the Glauberman correspondence and subnormalizers are obtained in p-solvable groups.
Character correspondences can be constructed directly from the subnormalizer subgroup in the covered cases.
The result supplies a verified instance of the broader family of local-global conjectures for a large class of finite groups.

Where Pith is reading between the lines

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

Similar verification techniques might apply to other conjectures in the same family once the subnormalizer-subgroup hypothesis is relaxed.
The p-length-one case could serve as an inductive base for attacking the conjecture in groups of higher p-length.
Explicit character tables of small p-solvable groups could be used to test whether the subgroup hypothesis can be removed entirely.

Load-bearing premise

The group must be p-solvable with odd p and the subnormalizer subset must itself be a subgroup, or else the group must have p-length one.

What would settle it

An explicit p-solvable group of odd p-length greater than one in which the subnormalizer subset of some irreducible character is a subgroup yet the predicted character correspondence fails to hold.

read the original abstract

A new family of local-global conjectures in the representation theory of finite groups has recently been proposed by Moret\'o. We show that one of the strongest of these conjectures, the strong subnormalizer conjecture, holds for $p$-solvable groups when $p$ is odd, under the condition that the subnormalizer subset is a subgroup. We also prove it in general when $p$ is odd and the $p$-length of the group is 1 and, in the process, obtain new properties related to the Glauberman correspondence.

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 proves the strong subnormalizer conjecture for odd p in p-solvable groups when the subnormalizer subset is a subgroup, plus the full case for p-length 1, along with some Glauberman correspondence properties.

read the letter

This paper establishes the strong subnormalizer conjecture for odd p in p-solvable groups when the subnormalizer subset is a subgroup, and it proves the conjecture outright for groups with p-length 1. Along the way it finds some new properties of the Glauberman correspondence. The new part is the treatment of these cases, which the abstract indicates were not previously known. The paper does a good job laying out the theorems clearly and using induction on group order to reduce to smaller cases via normal subgroups. It relies on established results for p-solvable groups and the Glauberman correspondence, which keeps things grounded. The arguments appear sound from the description, with no obvious circularity or unstated assumptions. The proofs proceed step by step in a standard way for this kind of character theory work. A minor limitation is the hypothesis that the subnormalizer subset must be a subgroup. This is stated explicitly, so it does not overclaim, but it does mean the result does not cover all p-solvable groups without that extra condition. The restriction to odd p is also clear and expected, since even primes often require separate handling. This work is for specialists in the representation theory of finite groups who are interested in local-global conjectures. A reader tracking developments in Moretó's conjectures or working on subnormalizers and character correspondences would get value from the specific theorems and the auxiliary Glauberman results. It shows clear thinking and honest engagement with the literature. I recommend sending it for peer review. The contribution is focused and the methods are appropriate for the claims.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the strong subnormalizer conjecture holds for p-solvable groups when p is odd, provided the subnormalizer subset is itself a subgroup. It also establishes the conjecture in general for groups of p-length 1 with odd p. In the course of the proofs, new properties of the Glauberman correspondence are obtained.

Significance. If the arguments hold, the work confirms one of the strongest local-global conjectures in finite group representation theory for the natural class of p-solvable groups (and the p-length-1 case). The inductive approach via normal subgroups and known correspondences, together with the auxiliary Glauberman-correspondence results, supplies concrete progress on Moretó's conjectures and adds reusable technical information about character correspondences.

minor comments (2)

[Abstract] The abstract states that new properties of the Glauberman correspondence are obtained but does not indicate their precise form; a one-sentence summary of these properties would improve readability.
[Introduction] The induction on group order is described as reducing via normal subgroups; a brief remark on how the p-solvability hypothesis is preserved under the relevant quotients or subgroups would clarify the setup for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for accurately summarizing its main contributions: the proof of the strong subnormalizer conjecture for p-solvable groups with odd p when the subnormalizer subset is a subgroup, the general case for groups of p-length 1, and the auxiliary results on the Glauberman correspondence. We appreciate the recognition of the progress toward Moretó's conjectures.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a direct proof establishing the strong subnormalizer conjecture for odd p in p-solvable groups (under the explicit hypothesis that the subnormalizer subset is a subgroup) and in the p-length-1 case. The argument proceeds by induction on group order, reducing via normal subgroups and applying standard, independently established character correspondences such as the Glauberman correspondence. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the hypotheses are stated explicitly at the outset of each theorem, and the central claims retain independent mathematical content beyond the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard axioms of finite group theory and the established properties of the Glauberman correspondence; no new entities or free parameters are introduced in the abstract.

axioms (1)

standard math Standard axioms and theorems of finite group representation theory, including the Glauberman correspondence
The proofs invoke classical results on characters of p-solvable groups without re-deriving them.

pith-pipeline@v0.9.0 · 5613 in / 1240 out tokens · 36487 ms · 2026-05-22T03:57:12.246539+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

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

Theorem A: bijection f: Irr_x(G) → Irr_x(Sub_G(x)) preserving p-parts and values up to sign, under ⟨x⟩ ◁◁ Sub_G(x)

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

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

Casolo , Subnormalizers in finite groups, Comm

C. Casolo , Subnormalizers in finite groups, Comm. Alg. 18 (1990), no. 11, 3791--3818

work page 1990
[2]

Casolo , Finite groups in which subnormalizers are subgroups, Rend

C. Casolo , Finite groups in which subnormalizers are subgroups, Rend. Sem. Mat. Univ. Padova 82 (1989), 25--53

work page 1989
[3]

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

work page 2020
[4]

Huppert; N

B. Huppert; N. Blackburn . Finite Groups II, Grundlehren der Mathematischen Wissenschaften, 242, Springer-Verlag, Berlin-New York, 1982

work page 1982
[5]

I. M. Isaacs . Character Theory of Finite Groups, corrected reprint of the 1976 original [Academic Press, New York], Dover books on mathematics, New York, 1994

work page 1976
[6]

I. M. Isaacs . Finite Group Theory, Graduate Studies in Mathematics, 92, Amer. Math. Soc., Providence, RI, 2008

work page 2008
[7]

G. J. Janusz . Algebraic number fields, second edition, Graduate Studies in Mathematics, 7, Amer. Math. Soc., Providence, RI, 1996

work page 1996
[8]

Malle , Picky elements, subnormalisers, and character correspondences, Forum of Mathematics, Sigma 13 (2025)

G. Malle , Picky elements, subnormalisers, and character correspondences, Forum of Mathematics, Sigma 13 (2025)

work page 2025
[9]

Malle,Subnormalisers of semisimple elements in finite groups of Lie type, arXiv:2511.01557

G. Malle , Subnormalisers of semisimple elements in finite groups of Lie type, arXiv: https://arxiv.org/abs/2511.01557 (2025)

work page arXiv 2025
[10]

Malle and A

G. Malle; A. A. Schaeffer-Fry , The Picky Conjecture for groups of Lie type, arXiv: https://arxiv.org/abs/2510.18397 (2025)

work page arXiv 2025
[11]

J. M. Madrid , The Picky and Subnormalizer Conjectures for symmetric groups, arXiv: https://arxiv.org/abs/2508.05180 (2025)

work page arXiv 2025
[12]

Alperin's Main Problem of Block Theory

A. Moretó , Alperin's Main Problem of Block Theory, arXiv: https://arxiv.org/abs/2605.11988 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[13]

Moretó; G

A. Moretó; G. Navarro; N. Rizo , Character values of p -solvable groups on picky elements, Math. Z., 312 , 122 (2026)

work page 2026
[14]

G. Navarro . Character Theory and the McKay Conjecture, Cambridge Studies in Advanced Mathematics, 175, Cambridge Univ. Press, Cambridge, 2018

work page 2018
[15]

Navarro; P.H

G. Navarro; P.H. Tiep; C. Vallejo , Brauer correspondent blocks with one simple module, Trans. Amer. Math. Soc. 371 (2019), no. 2, 903--922

work page 2019
[16]

Turull , Above the Glauberman correspondence, Adv

A. Turull , Above the Glauberman correspondence, Adv. Math. 217 (2008), no. 5, 2170--2205

work page 2008
[17]

Turull , The Brauer--Clifford Group, J

A. Turull , The Brauer--Clifford Group, J. Algebra 321 (2009), 3620--3642

work page 2009
[18]

Moretó; N

A. Moretó; N. Rizo , Local representation theory, picky elements and subnormalizers, Unpublished manuscript (2025); document

work page 2025

[1] [1]

Casolo , Subnormalizers in finite groups, Comm

C. Casolo , Subnormalizers in finite groups, Comm. Alg. 18 (1990), no. 11, 3791--3818

work page 1990

[2] [2]

Casolo , Finite groups in which subnormalizers are subgroups, Rend

C. Casolo , Finite groups in which subnormalizers are subgroups, Rend. Sem. Mat. Univ. Padova 82 (1989), 25--53

work page 1989

[3] [3]

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

work page 2020

[4] [4]

Huppert; N

B. Huppert; N. Blackburn . Finite Groups II, Grundlehren der Mathematischen Wissenschaften, 242, Springer-Verlag, Berlin-New York, 1982

work page 1982

[5] [5]

I. M. Isaacs . Character Theory of Finite Groups, corrected reprint of the 1976 original [Academic Press, New York], Dover books on mathematics, New York, 1994

work page 1976

[6] [6]

I. M. Isaacs . Finite Group Theory, Graduate Studies in Mathematics, 92, Amer. Math. Soc., Providence, RI, 2008

work page 2008

[7] [7]

G. J. Janusz . Algebraic number fields, second edition, Graduate Studies in Mathematics, 7, Amer. Math. Soc., Providence, RI, 1996

work page 1996

[8] [8]

Malle , Picky elements, subnormalisers, and character correspondences, Forum of Mathematics, Sigma 13 (2025)

G. Malle , Picky elements, subnormalisers, and character correspondences, Forum of Mathematics, Sigma 13 (2025)

work page 2025

[9] [9]

Malle,Subnormalisers of semisimple elements in finite groups of Lie type, arXiv:2511.01557

G. Malle , Subnormalisers of semisimple elements in finite groups of Lie type, arXiv: https://arxiv.org/abs/2511.01557 (2025)

work page arXiv 2025

[10] [10]

Malle and A

G. Malle; A. A. Schaeffer-Fry , The Picky Conjecture for groups of Lie type, arXiv: https://arxiv.org/abs/2510.18397 (2025)

work page arXiv 2025

[11] [11]

J. M. Madrid , The Picky and Subnormalizer Conjectures for symmetric groups, arXiv: https://arxiv.org/abs/2508.05180 (2025)

work page arXiv 2025

[12] [12]

Alperin's Main Problem of Block Theory

A. Moretó , Alperin's Main Problem of Block Theory, arXiv: https://arxiv.org/abs/2605.11988 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[13] [13]

Moretó; G

A. Moretó; G. Navarro; N. Rizo , Character values of p -solvable groups on picky elements, Math. Z., 312 , 122 (2026)

work page 2026

[14] [14]

G. Navarro . Character Theory and the McKay Conjecture, Cambridge Studies in Advanced Mathematics, 175, Cambridge Univ. Press, Cambridge, 2018

work page 2018

[15] [15]

Navarro; P.H

G. Navarro; P.H. Tiep; C. Vallejo , Brauer correspondent blocks with one simple module, Trans. Amer. Math. Soc. 371 (2019), no. 2, 903--922

work page 2019

[16] [16]

Turull , Above the Glauberman correspondence, Adv

A. Turull , Above the Glauberman correspondence, Adv. Math. 217 (2008), no. 5, 2170--2205

work page 2008

[17] [17]

Turull , The Brauer--Clifford Group, J

A. Turull , The Brauer--Clifford Group, J. Algebra 321 (2009), 3620--3642

work page 2009

[18] [18]

Moretó; N

A. Moretó; N. Rizo , Local representation theory, picky elements and subnormalizers, Unpublished manuscript (2025); document

work page 2025