arxiv: 2604.24565 · v1 · submitted 2026-04-27 · 🧮 math.RT · math.GR

Recognition: unknown

The Main Problem of Block Theory: Picky Elements and Subnormalizers

Alexander Moret\'o

Pith reviewed 2026-05-07 17:34 UTC · model grok-4.3

classification 🧮 math.RT math.GR MSC 20C20

keywords picky elementssubnormalizersblock theorylocal rulescharacter valuesp-elementsrepresentation theoryAlperin problem

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The pith

Sets Irr^x(G) and Sub_G(x) attached to p-elements are the natural objects for local rules on character values.

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

The paper presents a research programme in local representation theory that seeks rules determining character values using only local data from the group. It moves past earlier limits to characters of p-prime degree, height-zero characters, and blocks with abelian defect. The central focus is Alperin's main problem of block theory, the search for such local rules. Conjectures on picky elements and subnormalizers propose that the sets of certain irreducible characters and the subgroups linked to each p-element supply the right local information. A sympathetic reader would care because these objects could let one constrain or compute character values from local subgroup data alone.

Core claim

In the direction of Alperin's main problem, the conjectures on picky elements and subnormalizers suggest that the sets Irr^x(G) and the subgroups Sub_G(x) are the natural objects attached to a p-element x for the search for local rules for character values.

What carries the argument

The sets Irr^x(G) of irreducible characters and the subgroups Sub_G(x) attached to a given p-element x in the finite group G.

If this is right

Character values become subject to local restrictions based on the characters and subgroups tied to each p-element.
The study of blocks extends to all defect groups without requiring them to be abelian.
Local-global principles in representation theory gain a uniform description through these specific collections.
Progress on Alperin's main problem becomes possible by examining picky elements and their subnormalizers.

Where Pith is reading between the lines

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

These objects might allow verification of other local-global conjectures by checking consistency with known character tables in small groups.
Subnormal subgroups could receive renewed attention as organizing tools for representation-theoretic data.
The framework offers a route to predict values in groups too large for full character table computation.

Load-bearing premise

That local rules for character values exist and can be captured by the collections Irr^x(G) and Sub_G(x) rather than by classical invariants such as p'-degree or height zero.

What would settle it

An explicit finite group G and p-element x where the values of irreducible characters fail to follow any rule determined by Irr^x(G) and Sub_G(x) while still obeying known classical block invariants.

read the original abstract

This article is essentially an English translation of a paper of mine, published in \emph{La Gaceta de la RSME}. Its aim is to present, for a broad mathematical audience, a research programme in local representation theory that goes beyond the classical restrictions to characters of $p'$-degree, characters of height zero, and blocks of abelian defect. The final and most recent part of this programme concerns Alperin's main problem of block theory: the search for local rules for character values. In that direction I describe the conjectures on picky elements and subnormalizers, which suggest that the sets ${\rm Irr}^x(G)$ and the subgroups ${\rm Sub}_G(x)$ are the natural objects attached to a $p$-element $x$.

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 is an English translation of an earlier Spanish paper that sketches conjectures on picky elements and subnormalizers as possible new tools for local character rules, without proofs or checks.

read the letter

The core point is that this arXiv version adds nothing new mathematically. It is a translation of work Moretó already published in La Gaceta de la RSME, and the text simply describes a research programme aimed at local rules for character values that goes past the usual p'-degree or height-zero restrictions. The proposed objects are the sets Irr^x(G) and the subgroups Sub_G(x) attached to a p-element x, framed as the natural ones to study for Alperin's main problem in block theory. The writing is clear and the motivation is laid out plainly for a general audience. That is the main positive: it gives a concise statement of one person's view of a possible next direction in local representation theory. The soft spots are straightforward and central. There are no theorems, no examples worked through, no small-group calculations, and no attempt to show that these sets or subgroups actually produce new local-global information or improve on classical invariants. The conjectures are presented as a guiding intuition rather than something supported by evidence in the paper itself. Because the content is already out in another language, the translation does not change the evidential picture. This piece is mainly for people already deep in finite-group representation theory who follow open questions in block theory and want a short overview of one proposed programme. A reader outside that niche or looking for concrete results will not get much from it. I would not bring it to a reading group, as there is no specific claim or calculation to examine. I would not cite it in my own work in the next year, since it contains no new theorems or data. The thinking is coherent and engages honestly with the literature, so it qualifies as serious on its own terms. I would not recommend sending it to peer review. The material is already published elsewhere and remains purely conjectural and expository.

Referee Report

0 major / 1 minor

Summary. The manuscript is an expository English translation of a prior article in La Gaceta de la RSME. It outlines a research programme in the local representation theory of finite groups that seeks local rules for character values extending beyond the classical focus on p'-degree characters, height-zero characters, and blocks of abelian defect. The central part of the programme addresses Alperin's main problem of block theory; the author describes conjectures on picky elements and subnormalizers, proposing that the sets Irr^x(G) and the subgroups Sub_G(x) attached to a p-element x are the natural objects for formulating such local rules.

Significance. If the conjectures on picky elements and subnormalizers can be developed into precise statements with verifiable consequences, the programme could supply a new organizing principle for character values inside p-blocks and thereby contribute to Alperin's problem. The paper's immediate value is as a clear, accessible survey that frames open questions without overclaiming results. A strength is the explicit positioning of the material as a guiding intuition for future work rather than as established theorems.

minor comments (1)

Abstract: the sets Irr^x(G) and Sub_G(x) are introduced by name only; a single sentence recalling their definitions or pointing to the relevant earlier literature would help readers who encounter the programme for the first time.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and for the recommendation to accept. The referee's summary accurately captures the paper's purpose as an expository English translation of prior work in La Gaceta de la RSME, presenting a research programme in local representation theory that extends beyond classical restrictions and addresses Alperin's main problem via conjectures on picky elements and subnormalizers.

Circularity Check

0 steps flagged

Expository programme with no derivation chain

full rationale

The paper is explicitly an expository translation presenting a research programme and open conjectures on picky elements and subnormalizers. It contains no equations, fitted parameters, predictions, or first-principles derivations that reduce to their own inputs. The suggestion that Irr^x(G) and Sub_G(x) are natural objects is framed as guiding intuition for future work rather than a claim resting on any internal identity or computation. No self-citations or ansatzes are load-bearing in a derivation sense. The text is therefore self-contained against external benchmarks with no circularity present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are stated in the abstract; the terms picky elements and subnormalizers are introduced as new objects but without further specification of their definition or evidence.

pith-pipeline@v0.9.0 · 5420 in / 1098 out tokens · 73494 ms · 2026-05-07T17:34:52.743241+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 7 canonical work pages

[1]

J. L. Alperin , The main problem of block theory, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975), 341--356, Academic Press, New York--London, 1976

1975
[2]

J. L. Alperin , Local representation theory, The Santa Cruz Conference on Finite Groups (Santa Cruz, 1979), 369--375, Proc. Sympos. Pure Math. 37, American Mathematical Society, Providence, RI, 1980

1979
[3]

H. Blau, G. Michler , Modular representation theory of finite groups with T.I. Sylow p -subgroups, Trans. Amer. Math. Soc. 319 (1990), 417--468

1990
[4]

Broto , Sistemas de fusión en Álgebra y Topología, La Gaceta de la RSME 21 (2018), 183--202

C. Broto , Sistemas de fusión en Álgebra y Topología, La Gaceta de la RSME 21 (2018), 183--202

2018
[5]

Cabanes, B

M. Cabanes, B. Sp\"ath , The McKay conjecture on character degrees, Annals of Math. (2), to appear, arXiv:2410.20392

work page arXiv
[6]

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

1989
[7]

Casolo , Subnormalizers in finite groups, Comm

C. Casolo , Subnormalizers in finite groups, Comm. Algebra 18 (1990), 3791--3818

1990
[8]

Casolo , On the subnormalizer of a p -subgroup, J

C. Casolo , On the subnormalizer of a p -subgroup, J. Pure Appl. Algebra 77 (1992), 231--238

1992
[9]

Eaton, A

C. Eaton, A. Moret\'o , Extending Brauer's height zero conjecture to blocks with nonabelian defect groups, Int. Math. Res. Not. IMRN 20 (2014), 5581--5601

2014
[10]

The GAP Group , GAP --- Groups, Algorithms, and Programming, Version 4.12.2, 2022, http://www.gap-system.org

2022
[11]

Giannelli, J

E. Giannelli, J. M. Martínez, A. A. Schaeffer Fry , Character degrees in blocks and defect groups, J. Algebra 594 (2022), 170--193

2022
[12]

Giudici, L

M. Giudici, L. Morgan, C. Praeger , Finite permutation groups with quasi-semiregular elements, arXiv:2507.12866

work page arXiv
[13]

Gluck , Bounding the number of character degrees of a solvable group, J

D. Gluck , Bounding the number of character degrees of a solvable group, J. London Math. Soc. (2) 31 (1985), 457--462

1985
[14]

Goldstein, R

D. Goldstein, R. Guralnick, M. Lewis, A. Moret\'o, G. Navarro, P. H. Tiep , Groups with exactly one irreducible character of degree divisible by p , Algebra Number Theory 8 (2014), 397--428

2014
[15]

I. M. Isaacs , Character Theory of Finite Groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York--London, 1976

1976
[16]

I. M. Isaacs, A. Moret\'o, G. Navarro, P. H. Tiep , Groups with just one character degree divisible by a given prime, Trans. Amer. Math. Soc. 361 (2009), 6521--6547

2009
[17]

Keller , Orbit sizes and character degrees

T. Keller , Orbit sizes and character degrees. III, J. Reine Angew. Math. 545 (2002), 1--17

2002
[18]

Kessar, G

R. Kessar, G. Malle , Quasi-isolated blocks and Brauer's height zero conjecture, Annals of Math. (2) 178 (2013), 321--385

2013
[19]

Malle , Picky elements, subnormalisers, and character correspondences, Forum Math

G. Malle , Picky elements, subnormalisers, and character correspondences, Forum Math. Sigma 13 (2025), Paper No. e161, 21 pp

2025
[20]

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:2511.01557

work page arXiv
[21]

Malle, A

G. Malle, A. Moret\'o, N. Rizo , Minimal heights and defect groups with two character degrees, Adv. Math. 441 (2024), 22 pp

2024
[22]

Malle, G

G. Malle, G. Navarro, A. A. Schaeffer Fry, P. H. Tiep , Brauer's height zero conjecture, Annals of Math. (2) 200 (2024), 557--608

2024
[23]

Malle, A

G. Malle, A. A. Schaeffer Fry , On minimal positive heights for blocks of almost quasi-simple groups, arXiv:2410.22745

work page arXiv
[24]

Malle and A

G. Malle, A. A. Schaeffer Fry , The Picky Conjecture for groups of Lie type, arXiv:2510.18397

work page arXiv
[25]

A. Marcus , Alperin's weight conjecture and abelian defect groups, Proceedings of the conference on representation theory of finite groups and related topics, Cluj-Napoca, 2008, https://math.ubbcluj.ro/ marcus/papers/marcus_proc-Cluj08.pdf

2008
[26]

Mar\'oti, J

A. Mar\'oti, J. Martínez Madrid, A. Moret\'o , Covering the set of p -elements in finite groups by Sylow p -subgroups, J. Algebra 638 (2024), 840--861

2024
[27]

Mart´ ınez Madrid,The Picky and Subnormalizer Conjectures for sym- metric groups, arXiv:2508.05180

J. Martínez Madrid , The Picky and Subnormalizer Conjectures for symmetric groups, arXiv:2508.05180

work page arXiv
[28]

Moret\'o , Heights of characters and defect groups, Finite Groups 2003 (Gainesville, FL, 2003), 267--273, Walter de Gruyter, Berlin, 2004

A. Moret\'o , Heights of characters and defect groups, Finite Groups 2003 (Gainesville, FL, 2003), 267--273, Walter de Gruyter, Berlin, 2004

2003
[29]

Moret\'o , 25 years since I first met Gabriel, Valencia, 2025, https://www.uv.es/jomimar8/valenciaslides/moreto.pdf

A. Moret\'o , 25 years since I first met Gabriel, Valencia, 2025, https://www.uv.es/jomimar8/valenciaslides/moreto.pdf

2025
[30]

Moret\'o , El problema principal de la teoría de bloques: elementos picky y subnormalizadores, La Gaceta de la RSME 29 (2026), no

A. Moret\'o , El problema principal de la teoría de bloques: elementos picky y subnormalizadores, La Gaceta de la RSME 29 (2026), no. 1, 133--151, https://doi.org/10.63427/WJAH8548

work page doi:10.63427/wjah8548 2026
[31]

Moret\'o , Alperin's Main Problem of Block Theory, in preparation

A. Moret\'o , Alperin's Main Problem of Block Theory, in preparation
[32]

Moret\'o, G

A. Moret\'o, G. Navarro, N. Rizo , Character values of p -solvable groups on picky elements, Math. Z. 312 (2026), Article No. 122

2026
[33]

Moret\'o, N

A. Moret\'o, N. Rizo , Local representation theory, picky elements and subnormalizers, unpublished manuscript
[34]

Moret\'o, A

A. Moret\'o, A. A. Schaeffer Fry , A normal version of Brauer's height zero conjecture, Math. Ann. 395 (2026), Article No. 27

2026
[35]

Navarro , Characters and Blocks of Finite Groups, London Math

G. Navarro , Characters and Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge, 1998

1998
[36]

Navarro , The Eaton--Moretó conjecture and p -solvable groups, J

G. Navarro , The Eaton--Moretó conjecture and p -solvable groups, J. Algebra 665 (2025), 1--6

2025
[37]

Ruhstorfer , The Alperin--McKay and Brauer's height zero conjecture for the prime 2 , Annals of Math

L. Ruhstorfer , The Alperin--McKay and Brauer's height zero conjecture for the prime 2 , Annals of Math. (2) 201 (2025), 379--457

2025