Character degrees and local subgroups revisited

J. Miquel Mart\'inez

Character degrees and local subgroups revisited

T0 review · 0 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For q-solvable groups, the containment $\mathrm{Irr}_{p'}(G)\subseteq \mathrm{Irr}_{q'}(G)$ holds exactly when some Sylow $p$-subgroup $P$ and Sylow $q$-subgroup $Q$ satisfy $N_G(P)\subseteq N_G(Q)$ and $C_{Q'}(P)=1$.

desk verdict A clean, honest extension of Navarro-Wolf to q-solvable groups; the proofs are coherent but inherit load-bearing risk from two unrefereed deep preprints. read the letter →

arxiv 2411.15968 v2 pith:MVMDZGXI submitted 2024-11-24 math.GR math.RT

J. Miquel Martínez This is my paper · ORCID

classification math.GRmath.RT MSC 20C1520C2020D2020D10

keywords irreduciblecharacterscharacterdegreesq-solvablegroupsSylownormalizersq-blocksdefectMcKayconjectureGlaubermancorrespondence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a local criterion for a global containment of character degrees in finite $q$-solvable groups: the irreducible characters whose degree is not divisible by $p$ all have degree not divisible by $q$ exactly when a Sylow $p$-subgroup $P$ and a Sylow $q$-subgroup $Q$ can be chosen with $N_G(P)\subseteq N_G(Q)$ and $C_{Q'}(P)=1$. This removes the $p$-solvability condition from a theorem of Navarro and Wolf, which had been proved only for $\{p,q\}$-separable groups. The same paper proves the block version: if every character in a $q$-block $B$ has $p'$-degree, then some defect group of $B$ normalizes a Sylow $p$-subgroup of $G$. The proofs are powered by the recently proved McKay conjecture and by a degree-divisibility refinement of the McKay bijection, so the first theorem is explicitly conditional on that conjecture being true. Why it matters: two checks on Sylow subgroups and one centralizer now decide a statement about all irreducible degrees.

What carries the argument

The carrying device is a degree-divisibility-preserving McKay bijection $f:\mathrm{Irr}_{q'}(G)\to\mathrm{Irr}_{q'}(N_G(Q))$ with $f(\chi)(1)$ dividing $\chi(1)$ and $\chi(1)/f(\chi)(1)$ dividing $|G:N_G(Q)|$. Because $p$ does not divide $|G:N_G(Q)|$ once $N_G(P)\subseteq N_G(Q)$, this bijection lets the author compare $p'$-degree character counts in $G$ with those in the normalizer, turning the global inclusion into a local statement. In the local group, the Glauberman correspondence makes $C_{Q'}(P)=1$ equivalent to $P$ having a unique invariant irreducible character of $Q'$, and that uniqueness is what forces $p'$-degree characters to lie over the trivial character of $Q'$. For Theorem B, the same bijection is replaced by an induction with Fong–Reynolds reductions, character-triple isomorphisms, a classification-dependent relative $\pi'$-degree theorem for simple groups, and coprime Sylow action.

What would settle it

Compute, for a corpus of $q$-solvable groups and two primes $p\neq q$, whether $\mathrm{Irr}_{p'}(G)\subseteq\mathrm{Irr}_{q'}(G)$ holds while $N_G(P)\subseteq N_G(Q)$ and $C_{Q'}(P)=1$ fail for the chosen Sylow subgroups, or vice versa. A single violation would refute Theorem A; a direct search could start with solvable groups of small order, where character tables and Sylow normalizers are readily computed. Because Theorem A is conditional on the McKay bijection, a $q$-solvable group for which the degree-divisibility bijection of Theorem 1 does not exist would also settle the question.

Watch

Extended reading notes

Core claim

The central claim is Theorem A: in any finite $q$-solvable group $G$ and for distinct primes $p,q$, $\mathrm{Irr}_{p'}(G)\subseteq \mathrm{Irr}_{q'}(G)$ if and only if there exist $P\in\mathrm{Syl}_p(G)$ and $Q\in\mathrm{Syl}_q(G)$ such that $N_G(P)\subseteq N_G(Q)$ and $C_{Q'}(P)=1$. The forward direction produces the normalizer containment by passing through Brauer characters via Fong–Swan and invoking the $q$-solvable analogue of the Brauer-character result; the reverse direction uses the degree-divisibility McKay bijection and the Glauberman correspondence. Theorem B extends the same philosophy to $q$-blocks: if a $q$-block $B$ has no character whose degree is divisible by $p$, then some defect group of $B$ normalizes a Sylow $p$-subgroup. Both theorems extend results previously known only for $\{p,q\}$-separable groups, and both proofs rely on the classification of finite simple groups, with Theorem A depending on the truth of the McKay conjecture.

Load-bearing premise

The load-bearing premise is that the McKay conjecture is true as proved for $q$-solvable groups, and in particular that the quoted bijection between $\mathrm{Irr}_{q'}(G)$ and $\mathrm{Irr}_{q'}(N_G(Q))$ preserves divisibility of degrees; the paper states explicitly that if that conjecture were false, Theorem A would fail. Theorem B inherits a separate dependence on the classification-verified inductive McKay conditions for the simple groups used in the relative degree theorem.

Editorial extensions

If this is right

The Navarro–Wolf character-degree theorem now covers all $q$-solvable groups, not just groups that are simultaneously $p$-solvable and $q$-solvable.
A purely local check—two Sylow normalizers plus one centralizer—certifies whether the global inclusion $\mathrm{Irr}_{p'}(G)\subseteq\mathrm{Irr}_{q'}(G)$ holds.
For $q$-blocks, the $p'$-degree condition forces a defect group to sit inside a Sylow $p$-normalizer, a constraint that can obstruct the existence of such blocks.
The $\mathrm{PSL}(2,3^5)$ example from [NW98] shows $q$-solvability cannot simply be replaced by $p$-solvability, so the paper's hypothesis is close to optimal.
The proof confirms the earlier speculation that the McKay conjecture is the mechanism behind the stronger $q$-solvable version.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implicit consequence of the proof structure is that Theorem A is not self-contained: its validity is tied to the recent McKay theorem, so any future correction to that theorem would change the status of the characterization rather than just its proof.
The same degree-divisibility bijection could plausibly characterize other inclusions of prime-avoiding degree sets, such as $\mathrm{Irr}_{\pi'}(G)\subseteq\mathrm{Irr}_{\rho'}(G)$ for arbitrary sets of primes, once matching local conditions are formulated.
The paper leaves the height-zero version of Theorem B open; the Fong–Reynolds and relative $\pi'$-degree machinery used here is a direct starting point for that version.
A computational census of $q$-solvable groups (say $q=2$ or $3$) could test the sharpness of Theorem B for nonprincipal blocks, where the author reports finding no $p$-solvable counterexamples.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is exactly what it claims to be. It removes the p-solvability hypothesis from two Navarro-Wolf theorems by leaning on the recently proved McKay conjecture and a Liebeck-Navarro-Praeger-Tiep result. I read the internal arguments carefully and they hold together. The real question is the external dependence, and the author is admirably upfront about it.

What is new: Theorem A gives an iff for q-solvable groups: Irr_{p'}(G) is contained in Irr_{q'}(G) iff N_G(P) is contained in N_G(Q) and C_{Q'}(P)=1. Theorem B gives the block version. Both were known only for {p,q}-separable groups. The proof of Theorem A starts with a local reduction (Proposition 2) that uses the McKay bijection to move the degree condition to N_G(Q), then a Fong-Swan argument to compare q-Brauer characters and invokes [BNRS22]. The argument is clean, and I could not find a gap. Theorem B is an induction proof with character triple isomorphisms and Fong-Reynolds reductions; it relies on [LNPT24, Theorem A] in the final step. Again, the reduction is standard and the induction works.

The soft spot is exactly what the author says: Theorem A uses [CS24] to get |Irr_{p'}(G)|=|Irr_{p'}(N_G(P))| for a group that need not be p-solvable. That is a direct application of the McKay conjecture 'with character degrees' proved in that preprint. If [CS24] has a gap or does not cover this setting, Theorem A collapses. Theorem B depends on [LNPT24], which is submitted and itself depends on the inductive McKay conditions. These are not flaws in this paper's logic; they are honest dependencies on results that are not yet refereed. If I were the referee, I would ask the author to confirm that the statements in [CS24] and [LNPT24] really apply to q-solvable groups and to check whether those papers have been updated or accepted. Also, the citation to [NT] (in preparation) for an unconditional equality result is a minor irritant, not an error.

Who is this for? Finite group theorists working in character degrees, block theory, or the McKay conjecture. It is a solid, focused note. It deserves serious peer review; the referee's job will be to verify the external dependencies, not to pick apart the internal reductions.

Referee Report

0 major / 4 minor

Summary. The paper proves two theorems (Theorems A and B) that remove the p-solvability assumption from two results of Navarro and Wolf, replacing it with q-solvability. Theorem A characterizes, for a finite q-solvable group G, the inclusion Irr_p'(G) ⊆ Irr_q'(G) in terms of the local condition N_G(P) ⊆ N_G(Q) and C_Q'(P)=1 for suitable Sylow subgroups P and Q. Theorem B shows that if a q-block B of G has no character degree divisible by p, then some defect group of B normalizes a Sylow p-subgroup of G. The proofs use the McKay conjecture as proved in [CS24], the Fong-Swan theorem, a degree-divisibility refinement for q-solvable groups (Theorem 1, based on [Riz19] and [Gec20]), and a recent theorem of Liebeck, Navarro, Praeger, and Tiep [LNPT24].

Significance. This is a clean and concise paper that confirms a speculation of Navarro and Wolf. The main results are natural and improve the understanding of when p-degree restrictions on characters force local structural conditions. The author is transparent that the results depend on the recently proved McKay conjecture and on the theorem of [LNPT24], both of which are very recent and not yet fully refereed. The internal reductions are coherent and I found no local mathematical gap. If the external results hold, the paper is a solid contribution to the character theory of q-solvable groups.

minor comments (4)

[Section 2, proof of Theorem A] The notation IBr_p'(G) is used for the set of q-Brauer characters of p'-degree. This is nonstandard and easily confused with the usual p-Brauer characters; please define it explicitly or use a clearer notation such as IBr_q(G)_{p'}.
[Section 3, proof of Theorem B] There are two occurrences of the typo 'Propostion' where 'Proposition' is intended.
[Section 3, proof of Theorem B, Step 1] The phrase 'arguing as before by for the prime q' contains a typo ('by for' should be 'for').
[Section 2, Proposition 2] The notation Irr_P(Q') for the set of P-invariant characters is standard, but since it appears without definition, a brief parenthetical explanation would help the reader.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: The two main theorems rest on external results ([CS24], [BNRS22], [LNPT24], [Nav18]) and the sole same-author citation [MR23] is a published technical tool, not a restatement of the target claims.

full rationale

The derivation chain for Theorem A is not circular. The forward direction in Proposition 2 uses Theorem 1 (a McKay correspondence for q-solvable groups with degree divisibility, proved via [Tur07], [Riz19], [Gec20], [HT94]) and the McKay conjecture as proved in [CS24] to compare cardinalities of Irr_{p'} and Irr_{pi'}; the reverse direction is handled by the Glauberman correspondence. The implication from character inclusion to local inclusion is then obtained by Fong–Swan and by [BNRS22, Theorem A], which is an external published result on Brauer character degrees. None of these inputs is the statement of Theorem A itself, and no parameter is fitted or defined in terms of the conclusion. Theorem B likewise proceeds by an induction whose external inputs are [LNPT24, Theorem 3], [Nav18], [NW01], and a Fong–Reynolds reduction. The paper's only self-citation is [MR23, Proposition 1.3] in the proof of Theorem B, where it supplies a (B,Q)-good character and the corresponding induction bijection; this is a published journal result used as a black-box technical tool, not a hidden assumption of Theorem B. The author explicitly discloses the load-bearing dependencies in the introduction: “while Theorem A depends on the McKay conjecture being true, Theorem B depends on the McKay conjecture being proved in a certain way, that is, proving that the inductive McKay conditions defined in [IMN07] hold for certain simple groups (we do not need this explicitly here but it is used in the proof of [LNPT24, Theorem A], on which our work relies)”. Dependence on unrefereed preprints is a real correctness or verification risk, but it is not circularity: those preprints are external to this paper, authored by other researchers, and the paper makes no attempt to derive them from its own results. The internal reductions (the cardinality argument in Proposition 2, the induction steps in Theorem 4, and the Fong–Reynolds descent in the proof of Theorem B) are coherent and do not reduce to the statements being proved. Accordingly, the circularity score is low; the value 2 reflects the presence of one minor self-citation that is not load-bearing in the circularity sense.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's central claims rest almost entirely on external deep results: the McKay conjecture [CS24], the classification, [BNRS22], and [LNPT24]. The only self-citation is [MR23] for a technical block-theoretic reduction. There are no free parameters or invented entities. The novelty is in the reduction and combination of these results, not in new axioms.

assumptions (7)

domain assumption The McKay conjecture holds as proved in [CS24]: for any finite group and any prime, the number of irreducible characters of degree coprime to that prime equals the same number for the normalizer of a Sylow subgroup.
Used in Proposition 2 to equate |Irr_{p'}(G)| with |Irr_{p'}(N_G(Q))|, a key step in proving the if and only if in Theorem A. The paper states 'Theorem A depends on the McKay conjecture being true'.
domain assumption Classification of finite simple groups (CFSG).
Invoked via [BNRS22, Theorem A], [LNPT24, Theorem 3], and Theorem 1 (through [Gec20] and [HT94]). The paper notes both [BNRS22, Theorem A] and Theorem A rely on CFSG.
domain assumption [BNRS22, Theorem A]: For q-solvable groups, IBr_{p'}(G) ⊆ IBr_{q'}(G) (as q-Brauer characters) if and only if N_G(P) ⊆ N_G(Q) for some Sylow p-subgroup P and Sylow q-subgroup Q.
Used in the proof of Theorem A to translate the Brauer character inclusion into the local condition, after Fong-Swan reduction.
domain assumption [LNPT24, Theorem 3] (Liebeck-Navarro-Praeger-Tiep): Under certain conditions on a normal subgroup Z and character λ, G/Z has abelian Hall π-subgroups.
Used in the final step of Theorem 4 to conclude that G/Z has abelian Hall π-subgroups, which yields the desired Sylow normalizer property. This result itself relies on CFSG and is from a submitted paper.
domain assumption [MR23, Proposition 1.3]: Existence of (B,Q)-good characters for a q-block B and defect group Q.
Used in the proof of Theorem B to either reduce to a proper subgroup or to a G-invariant character of O_{q'}(G). This is a self-citation, but a published technical result.
domain assumption [NW01, Theorem 3.1] and [NW01, Lemma 2.1]: Structural results used in the induction steps of Theorem 4.
Used in Steps 1 and 2 of Theorem 4 to handle p-special characters and O_{q'}(G/Z) reduction. These are classical results from the 2001 Navarro-Wolf paper.
standard math Standard character theory results: Fong-Swan theorem, Glauberman correspondence, Ito's theorem, Hall-Higman 1.2.3, Clifford correspondence, and character triple isomorphisms.
Used throughout as black-box results; they are standard in the field and cited appropriately (e.g., [Nav98], [Nav18], [Isa08], [Asc86]).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Character degrees and local subgroups revisited." pith.science (2026). https://pith.science/paper/MVMDZGXI

@misc{pith2026241115968,
  author       = {Pith},
  title        = {Pith review of: Character degrees and local subgroups revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MVMDZGXI}},
  note         = {Machine review of arXiv:2411.15968}
}

abstract

Let $p$ and $q$ be different primes and let $G$ be a finite $q$-solvable group. We prove that $\mathrm{Irr}_{p'}(G)\subseteq \mathrm{Irr}_{q'}(G)$ if and only if $\mathbf{N}_G(P)\subseteq \mathbf{N}_G(Q)$ and $\mathbf{C}_{Q'}(P)=1$ for some $P\in\mathrm{Syl}_p(G)$ and $Q\in\mathrm{Syl}_q(G)$. Further, if $B$ is a $q$-block of $G$ and $p$ does not divide the degree of any character in $\mathrm{Irr}(B)$ then we prove that a Sylow $p$-subgroup of $G$ is normalized by a defect group of $B$. This removes the $p$-solvability condition of two theorems of G. Navarro and T. R. Wolf.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 22 canonical work pages

[1]

M. G. Aschbacher, Finite group theory , Cambridge Studies in Advanced Mathematics, 10, Cambridge Univ. Press, Cambridge, 1986

work page 1986
[2]

Beltr\' a n and G

A. Beltr\' a n and G. Navarro. Sylow normalizers and B rauer character degrees. J. Algebra , 229(2):623--631, 2000

work page 2000
[3]

Bonazzi, G

L. Bonazzi, G. Navarro, N. Rizo, and L. Sanus. Brauer character degrees and S ylow normalizers. Ann. Mat. Pura Appl. (4) , 201(6):2575--2582, 2022

work page 2022
[4]

Cabanes and B

M. Cabanes and B. Sp \"a th. T he M c K ay conjecture con character degrees. arXiv:2410.20392 https://www.arxiv.org/abs/2410.20392, 2024

arXiv 2024
[5]

M. Geck. Green functions and G lauberman degree-divisibility. Ann. of Math. (2) , 192(1):229--249, 2020

work page 2020
[6]

Giannelli, G

E. Giannelli, G. Malle and C. Vallejo. E ven degree characters in principal blocks. J. Pure Appl. Algebra , 223:900--907, 2019

work page 2019
[7]

Giannelli and N

E. Giannelli and N. Mecacci. Even degree characters in blocks of symmetric and alternating groups . submitted , 2024

work page 2024
[8]

Hartley and A

B. Hartley and A. Turull. On characters of coprime operator groups and the G lauberman character correspondence. J. Reine Angew. Math. , 451:175--219, 1994

work page 1994

Show all 24 references

[9]

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

2008
[10]

I. M. Isaacs, G. Malle, and G. Navarro. A reduction theorem for the M c K ay conjecture. Invent. Math. , 170(1):33--101, 2007

2007
[11]

Kurzweil and B

H. Kurzweil and B. Stellmacher, The theory of finite groups , translated from the 1998 German original, Universitext, Springer, New York, 2004

1998
[12]

M. W. Liebeck, G. Navarro, C. E. Praeger, and P. H. Tiep. Characters of relative ' -degree over normal subgroups. submitted , 2024

2024
[13]

Y. Liu, L. Wang, W. Willems and J. Zhang. Brauer's height zero conjecture for two primes holds true. Math. Ann. 388:1677--1690, 2024

2024
[14]

Malle and G

G. Malle and G. Navarro. Brauer's height zero conjecture for two primes. Math. Z. , 295:1723--1732, 2020

2020
[15]

J. M. Mart\' nez and D. Rossi. Degree divisibility in A lperin- M c K ay correspondences. J. Pure Appl. Algebra , 227(12):Paper No. 107449, 2023

2023
[16]

Moret\'o and A

A. Moret\'o and A. A. Schaeffer Fry A normal version of Brauer's height zero conjecture. arXiv:2406.06428 https://arxiv.org/abs/2406.06428, 2024

2024 arXiv
[17]

G. Navarro. Characters and blocks of finite groups , volume 250 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 1998

1998
[18]

G. Navarro. Character theory and the M c K ay conjecture , volume 175 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2018

2018
[19]

Navarro, N

G. Navarro, N. Rizo, and L. Sanus. Character degrees in separable groups. Proc. Amer. Math. Soc. , 150(6):2323--2329, 2022

2022
[20]

Navarro, P

G. Navarro, P. H. Tiep. Characters, degrees, and sets of primes. in preparation
[21]

Navarro and T

G. Navarro and T. R. Wolf. Character degrees and local subgroups of -separable groups. Proc. Amer. Math. Soc. , 126(9):2599--2605, 1998

1998
[22]

Navarro and T

G. Navarro and T. R. Wolf. Character degrees and blocks of finite groups. J. Reine Angew. Math. , 531:141--146, 2001

2001
[23]

N. Rizo. Divisibility of degrees in M c K ay correspondences. Arch. Math. (Basel) , 112(1):5--11, 2019

2019
[24]

A. Turull. Degree divisibility in character correspondences. J. Algebra , 307(1):300--305, 2007

2007

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

M. G. Aschbacher, Finite group theory , Cambridge Studies in Advanced Mathematics, 10, Cambridge Univ. Press, Cambridge, 1986

work page 1986

[2] [2]

Beltr\' a n and G

A. Beltr\' a n and G. Navarro. Sylow normalizers and B rauer character degrees. J. Algebra , 229(2):623--631, 2000

work page 2000

[3] [3]

Bonazzi, G

L. Bonazzi, G. Navarro, N. Rizo, and L. Sanus. Brauer character degrees and S ylow normalizers. Ann. Mat. Pura Appl. (4) , 201(6):2575--2582, 2022

work page 2022

[4] [4]

Cabanes and B

M. Cabanes and B. Sp \"a th. T he M c K ay conjecture con character degrees. arXiv:2410.20392 https://www.arxiv.org/abs/2410.20392, 2024

arXiv 2024

[5] [5]

M. Geck. Green functions and G lauberman degree-divisibility. Ann. of Math. (2) , 192(1):229--249, 2020

work page 2020

[6] [6]

Giannelli, G

E. Giannelli, G. Malle and C. Vallejo. E ven degree characters in principal blocks. J. Pure Appl. Algebra , 223:900--907, 2019

work page 2019

[7] [7]

Giannelli and N

E. Giannelli and N. Mecacci. Even degree characters in blocks of symmetric and alternating groups . submitted , 2024

work page 2024

[8] [8]

Hartley and A

B. Hartley and A. Turull. On characters of coprime operator groups and the G lauberman character correspondence. J. Reine Angew. Math. , 451:175--219, 1994

work page 1994