Character Values of $p$-solvable groups on picky elements

Alexander Moret\'o; Gabriel Navarro; Noelia Rizo

REVIEW 3 major objections 3 minor 1 cited by

Character Values of $p$-solvable groups on picky elements

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a p-solvable group G with p odd and a picky p-element x, the paper constructs a bijection Irr_x(G) to Irr_x(N_G(P)) preserving p-parts and fixing values up to sign.

desk verdict The main theorem is plausible, but the proof of Theorem A as written relies on a normality assertion for KP that fails in a concrete p-solvable group, so the central reduction does not go through. read the letter →

arxiv 2506.11670 v1 pith:OLHRLYXD submitted 2025-06-13 math.RT math.GR

Alexander Moretó , Gabriel Navarro , Noelia Rizo This is my paper · ORCID

classification math.RTmath.GR MSC 20C15

keywords pickyelementsp-solvablegroupsMcKayconjectureGlaubermancorrespondencecharactertripleisomorphismsvaluesdegreeslocal-to-globalcorrespondences

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

This paper proves a conjecture about picky $p$-elements, meaning $p$-elements that lie in exactly one Sylow $p$-subgroup, for finite $p$-solvable groups with $p$ odd. The main theorem states that for such a group $G$, a picky $p$-element $x$, and the unique Sylow $p$-subgroup $P$ containing $x$, there is a bijection between the irreducible characters of $G$ that do not vanish at $x$ and the corresponding characters of $\mathrm{N}_G(P)$. The bijection preserves the $p$-part of the character degree and changes the character value at $x$ only by a single sign $\pm1$. This is a local-to-global statement in the spirit of the McKay conjecture, because it pins down a global character value and degree $p$-part using only the normalizer of a Sylow subgroup. The paper also explains why $p=2$ is different, giving a symplectic example where the sign attached to $2$-elements of order at least $4$ changes between elements of the same order.

What carries the argument

The engine is the relative Glauberman correspondence, stated as Theorem 2.2: if a $p$-group $P$ acts on a group $G$ and $G/N$ has order coprime to $p$, there is a natural bijection between $P$-invariant irreducible characters of $G$ and of $C_{G/N}(P)$, with the congruence $\chi_C=e\varphi+p\Delta+\Xi$ and $e\equiv\pm1\pmod p$. The sign $\varepsilon$ in Theorem A is essentially this residue $e$ read modulo $p$. The picky hypothesis ensures that $C_{K/L}(x)=C/L$, so the centralizer appearing in the Glauberman correspondence is the correct one. Character triple isomorphisms then transport the statement to the case where the relevant normal subgroup is central, and an induction on $|G:L|$, controlled by the Clifford correspondence, extends the result to the full group.

What would settle it

Take a $p$-solvable group with a picky $p$-element whose final reduction lands on a $p$-chief factor, compute the bijection from $\mathrm{Irr}_x(G)$ to $\mathrm{Irr}_x(\mathrm{N}_G(P))$, and compare the sign of the ratio for each irreducible character. If two characters give opposite signs, Theorem A fails; a computer search over all small $p$-solvable groups with odd $p$ would decide whether such a pair exists.

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Extended reading notes

Core claim

The paper's central claim, Theorem A, is that for an odd prime $p$, a finite $p$-solvable group $G$, and a picky $p$-element $x$ with unique Sylow $p$-subgroup $P$, there exists a bijection $*:\mathrm{Irr}_x(G)\to\mathrm{Irr}_x(\mathrm{N}_G(P))$ and a sign $\varepsilon_x$ such that $\chi(x)=\varepsilon_x\chi^*(x)$ and $\chi(1)_p=\chi^*(1)_p$ for every $\chi\in\mathrm{Irr}_x(G)$. The bijection is assembled from the relative Glauberman correspondence, which relates $x$-invariant characters of a normal subgroup to characters of a centralizer, and from character triple isomorphisms, which preserve ratios of character degrees and character values up to the relevant sign. The proof reduces to the case where the relevant normal quotient $K/L$ has order coprime to $p$, where the Glauberman congruence $[\theta_C,\varphi]\equiv\varepsilon\pmod p$ forces the sign, and then lifts the result through Clifford correspondents to the whole group.

Load-bearing premise

The final step of the proof is only valid when the normal subgroup factor that the reduction chooses has size not divisible by $p$; the case where that factor is a $p$-group is not separately handled.

Editorial extensions

If this is right

If Theorem A is correct, the picky-element conjecture holds for every finite $p$-solvable group with odd $p$, with the bijection preserving the $p$-part of each individual character degree.
Because the value at $x$ is preserved up to a single sign, the statement is strictly stronger than a McKay-style count: it controls the actual character values in the global group, not just the number of characters.
The final-section example with a symplectic group acting on an extraspecial 3-group implies that any $p=2$ version of the conjecture must allow the sign to depend on the element, at least for 2-elements of order 4 or more.
The reduction template, combining the relative Glauberman correspondence with character triple isomorphisms, gives a reusable route for proving local character correspondences in $p$-solvable groups.

Reading between the lines

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

The text leaves the $p$-chief-factor case untreated: the final reduction chooses $K/L$ of order coprime to $p$, and when $K/L$ is itself a $p$-group the cited key theorems do not directly apply.
The proof also appears to assume that the signs $\varepsilon_\theta$ obtained from different Clifford correspondents are all equal, since Theorem A needs one global sign $\varepsilon_x$ rather than a sign that varies with the character.
A natural test is to compute the bijection explicitly for a small $p$-solvable group where the final reduction lands on a $p$-chief factor and compare the signs; a single pair of characters with opposite signs would pinpoint exactly where the proof would need repair.
If the missing $p$-chief-factor case can be handled, the same induction may extend the theorem to all $p$-solvable groups without a new idea, since the obstruction appears structural rather than numerical.

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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

The main theorem is plausible, and the early parts of the paper are serious and well-argued, but the proof of Theorem A has a false step in the final reduction. The paper proves a McKay-type bijection for p-solvable groups with odd p, preserving values on picky elements and p-parts of degrees, and it is the first proof of the authors' recent conjecture in this class. Theorems 3.1-3.4 and 4.2 are substantial and use the relative Glauberman correspondence and Dade-Turull character triple isomorphisms correctly as far as I can tell. The discussion of the p=2 obstacle is honest and specific.

The problem is Theorem 4.5. The proof sets K=O_{p'p}(G) and asserts that KP is normal in G. That assertion is false in general. Take p=3 and G=C_3^3 ⋊ A_4, where A_4 acts irreducibly on C_3^3. Then O_{3'}(G)=1, so K=O_{3'3}(G)=C_3^3. If x is a 3-cycle in the A_4 complement, x is picky in G and the unique Sylow 3-subgroup P containing x is K⟨x⟩. But P is not normal in G, because its image in G/K≅A_4 is a nonnormal Sylow 3-subgroup. Hence KP=P is not normal, G≠K N_G(P), and the induction cannot apply Theorem 4.2. This is the same example that makes K/L a p-group, so the reader's p-chief-factor concern is real but is a symptom of the same bad choice of K. The sign-uniformity issue across Clifford correspondents is minor in comparison and likely repairable.

The intermediate theorems are not affected by this gap. The theorem itself may well be true, but as written the paper does not establish it. It needs a different reduction or an extra case analysis for chief factors inside O_{p'p}(G) that are p-groups.

This is a paper for specialists in character theory and the McKay program. It deserves a serious referee because the result is important and most of the proof is careful, but I would not accept it in current form. The referee should be asked to check the normality assertion in Theorem 4.5 and the p-chief-factor case. If the authors fix that, the paper would be a solid contribution.

Referee Report

3 major / 3 minor

Summary. The paper proves Theorem A: for a finite p-solvable group G with p odd and a picky p-element x (i.e., an element lying in a unique Sylow p-subgroup P), there exists a bijection Irr_x(G) -> Irr_x(N_G(P)) that preserves the p-parts of character degrees and changes character values only by a global sign. The proof proceeds by reduction to normal subgroups, using the relative Glauberman correspondence of Navarro–Tiep–Vallejo, character triple isomorphisms, and an induction on O_{p'p}(G). The main theorems leading to Theorem A are Theorems 3.1, 3.2, 3.4, and 4.2.

Significance. If the proof were correct, Theorem A would be a substantial step toward a McKay-type conjecture for picky elements, establishing the conjecture for all p-solvable groups with p odd and providing a template for further reductions. The paper draws appropriately on deep published results, including the relative Glauberman correspondence and Dade–Turull theory, and the overall strategy is attractive. However, the final reduction in the proof of Theorem A contains load-bearing gaps: the asserted normality of PK is false in general, the p-chief-factor case is not covered by the cited theorems, and the independence of the sign across the Clifford decomposition is not justified. These issues prevent the paper from establishing its central claim as written.

major comments (3)

[Theorem 4.5, proof] The proof asserts 'Notice that KP⊴G' without justification, but this is false in general. For p=3, take K=C_3^3 and let A=A_4 act on K via an irreducible 3-dimensional module over F_3; set G=K⋊A. Then O_{3'}(G)=1 and O_3(G)=K, so K=O_{3'3}(G). If x is a 3-cycle in A, then x is picky in G with unique Sylow 3-subgroup P=K⟨x⟩, yet P is not normal in G because its image in G/K≅A_4 is a nonnormal Sylow 3-subgroup. Hence PK=P is not normal. Since Theorem 4.2 and Theorem 3.4 require V=PK to be normal in G, the reduction stated in Theorem 4.5 collapses.
[Theorem 4.5, proof] The proof chooses a chief factor K/L of G below K=O_{p'p}(G) and applies Theorem 4.2, but Theorem 4.2 is stated only when K/L is a p'-group. When K/L is a p-group, which occurs in the example above with L=1, Theorem 4.2 does not apply. No separate treatment of the p-chief-factor case is given, so the induction step is incomplete.
[Theorem 4.5, final paragraph; Theorem 4.2] The bijection in Theorem A is assembled from the bijections Irr(G|θ)→Irr(H|φ) for θ in a set Δ of H-representatives of x-invariant characters of Irr(K). Theorem 4.2 supplies a sign ε_x for each θ, defined by [θ_C, φ]≡ε mod p. The proof does not show that this sign is independent of the choice of θ. Without such an argument one only obtains χ(x)=ε_θ χ*(x) on each piece Irr_x(G|θ), not a single ε_x for all χ∈Irr_x(G). This is load-bearing because the statement of Theorem A asserts a global sign.

minor comments (3)

[Theorem 4.5, proof] In the proof, the expression 'KP=O_{p'}(G)' appears to be a typo; it should likely be O_{p'p}(G) or the intended equality needs clarification.
[Section 4.5, notation] The notation Irr_x(G|θ) is used without definition; it should be defined as Irr_x(G)∩Irr(G|θ).
[Final remarks] The statement 'In Theorem 3.4, we saw that the sign associated to picky elements was universal' is potentially misleading: the sign in Theorem 3.4 is defined for a fixed θ and φ, and universality across different θ is exactly what is missing from the proof of Theorem A.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem A is proved from independent external results, not from the conjecture it establishes.

full rationale

The conjecture in [6] is the target of the proof, not an input assumption: the abstract and introduction state that the conjecture was proposed by the first and third authors, and the paper then proves it for p-solvable groups using published machinery. The load-bearing tools are the relative Glauberman correspondence of Navarro–Tiep–Vallejo [8], Dade–Turull and character-triple results from [7] and [9], and standard Isaacs character theory [2]. These are external, published results that do not contain Theorem A and are not reduced to it. The authorship overlaps in [6], [7], and [8] do not make the argument circular: [6] is only the source of the conjecture being proved, while [7] and [8] are independent published theorems. The sign epsilon_x is defined by the congruence [theta_C, phi] congruent to epsilon mod p and then derived for chi(x) through the relative Glauberman correspondence, so it is not a fitted parameter renamed as a prediction. There is a genuine rigor gap in the final reduction: in the proof of Theorem 4.5, the assertion 'Notice that KP ▷ G' is not justified, and a chief factor K/L of G with K=O_{p'p}(G) may be a p-group, whereas Theorems 3.4 and 4.2 are stated only for the case where K/L is a p'-group. The external counterexample (C_3^3 ⋊ A_4 with p=3) illustrates this failure. However, this is a correctness or hypothesis-verification problem, not circularity: no equation or construction in the proof is equivalent to the conclusion by definition, and no prediction is forced by a parameter fitted to the data it is said to predict. The explicit p=2 limitation in the final section is an acknowledged scope restriction, not a circular step. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numerical parameters are fitted; the proof is purely structural. The listed axioms are the deep character-theoretic tools the proofs rest on, all cited from published sources. No new particles, forces, or non-mathematical entities are introduced.

assumptions (5)

domain assumption Relative Glauberman correspondence (Theorem E of Navarro-Tiep-Vallejo [8]) provides a bijection between P-invariant irreducible characters of G and of the fixed-point section, with congruence conditions on restrictions (Theorem 2.2 in this paper).
Stated as Theorem 2.2 and used throughout: Lemma 2.3, Theorems 3.1, 3.4, and 4.2 all depend on this correspondence.
domain assumption Dade-Turull character correspondence and character triple isomorphisms (Navarro [7, Ch. 8.3], Turull [9]) preserve ratios of character degrees and map canonical extensions appropriately.
Invoked in Theorems 3.1 and 3.2 to construct the bijection between Irr(G|θ) and Irr(H|φ) with χ(1)_p = χ*(1)_p.
standard math Clifford theory facts about invariant characters, stabilizers, and Clifford correspondents (Isaacs [2], Wolf [10]).
Used in Lemma 2.1 and throughout the induction step in Theorem 3.4, including the claim that all constituents of θ_L are K-conjugate.
domain assumption Canonical extensions of invariant characters to groups with p-group quotients exist and are unique in the relevant setting (Gallagher; proof of Lemma 5.17 in [7]).
Theorem 3.1 relies on canonical extensions hat(θ_{p'}) and hat(φ_{p'}), and the congruence arguments use the associated character triple isomorphisms.
standard math In a p-solvable group, the Frattini argument gives G = K N_G(P) for a normal subgroup K containing P, and the structural properties of O_{p'p}(G) described in the proof of Theorem 4.5.
Used in Theorem 4.5 to set up the induction and define the subgroup H = L N_G(P).

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Cite this review

Pith. "Pith review of Character Values of $p$-solvable groups on picky elements." pith.science (2026). https://pith.science/paper/OLHRLYXD

@misc{pith2026250611670,
  author       = {Pith},
  title        = {Pith review of: Character Values of $p$-solvable groups on picky elements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLHRLYXD}},
  note         = {Machine review of arXiv:2506.11670}
}

abstract

A new conjecture on characters of finite groups, related to the McKay conjecture, was proposed recently by the first and third authors. In this paper, we prove it for $p$-solvable groups when $p$ is odd.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Finite permutation groups with quasi-semiregular elements
math.GR 2025-07 conditional novelty 6.0 of 10

Finite primitive permutation groups admitting a quasi-semiregular element have O'Nan-Scott type HA, AS, PA, SD or CD, and the alternating and sporadic almost simple cases are classified explicitly.

Reference graph

Works this paper leans on

10 extracted references · 9 canonical work pages · cited by 1 Pith paper

[1]

I. M. Isaacs, Characters of solvable and symplectic groups.Amer. J. Math.95(1973), 594–635

work page 1973
[2]

I. M. Isaacs,Character Theory of Finite Groups. Academic Press, New York-London, 1976

work page 1976
[3]

Malle, Picky elements, subnormalizers, and character correspondences

G. Malle, Picky elements, subnormalizers, and character correspondences. arXiv:2503.10425

arXiv
[4]

Mar´ oti, J

A. Mar´ oti, J. Mart ´ ınez Madrid and A. Moret´ o, Covering the set ofp-elements in finite groups by Sylowp-subgroups.J. Algebra638(2024), 840–861

work page 2024
[5]

Mart ´ ınez Madrid, Picky and subnormalizer conjectures in symmetric groups

J. Mart ´ ınez Madrid, Picky and subnormalizer conjectures in symmetric groups. In preparation

work page
[6]

Moret´ o and N

A. Moret´ o and N. Rizo, Local representation theory, picky elements and subnormalizers. In preparation

work page
[7]

Navarro,Character theory and the McKay conjecture

G. Navarro,Character theory and the McKay conjecture. Cambridge University Press, Cam- bridge, 2018

work page 2018
[8]

Navarro, P

G. Navarro, P. H. Tiep and C. Vallejo, Brauer correspondent blocks with one simple module. Trans. Amer. Math. Soc.371(2019), 903–922. 14 ALEXANDER MORET ´O, GABRIEL NA V ARRO, AND NOELIA RIZO

work page 2019

Show all 10 references

[9]

Turull, Above the Glauberman correspondence,Adv

A. Turull, Above the Glauberman correspondence,Adv. Math.217(2008), 2170–2205

2008
[10]

T. R. Wolf, Character correspondences induced by subgroups of operator groups,J. Algebra. 57(1979), 502–521. Departament de Matem`atiques, Universitat de V al`encia, 46100 Burjassot, V al`encia, Spain Email address:alexander.moreto@uv.es Departament de Matem`atiques, Universit...

1979

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

I. M. Isaacs, Characters of solvable and symplectic groups.Amer. J. Math.95(1973), 594–635

work page 1973

[2] [2]

I. M. Isaacs,Character Theory of Finite Groups. Academic Press, New York-London, 1976

work page 1976

[3] [3]

Malle, Picky elements, subnormalizers, and character correspondences

G. Malle, Picky elements, subnormalizers, and character correspondences. arXiv:2503.10425

arXiv

[4] [4]

Mar´ oti, J

A. Mar´ oti, J. Mart ´ ınez Madrid and A. Moret´ o, Covering the set ofp-elements in finite groups by Sylowp-subgroups.J. Algebra638(2024), 840–861

work page 2024

[5] [5]

Mart ´ ınez Madrid, Picky and subnormalizer conjectures in symmetric groups

J. Mart ´ ınez Madrid, Picky and subnormalizer conjectures in symmetric groups. In preparation

work page

[6] [6]

Moret´ o and N

A. Moret´ o and N. Rizo, Local representation theory, picky elements and subnormalizers. In preparation

work page

[7] [7]

Navarro,Character theory and the McKay conjecture

G. Navarro,Character theory and the McKay conjecture. Cambridge University Press, Cam- bridge, 2018

work page 2018

[8] [8]

Navarro, P

G. Navarro, P. H. Tiep and C. Vallejo, Brauer correspondent blocks with one simple module. Trans. Amer. Math. Soc.371(2019), 903–922. 14 ALEXANDER MORET ´O, GABRIEL NA V ARRO, AND NOELIA RIZO

work page 2019