The McKay conjecture with coprime group automorphisms and the Okuyama-Wajima argument

Adele Maltempo; Carolina Vallejo

REVIEW 1 major objections 4 minor 19 references

The McKay conjecture with coprime group automorphisms and the Okuyama-Wajima argument

T0 review · 1 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper proves that in p-solvable groups the McKay bijection — between characters of degree not divisible by p and characters of the normalizer of a Sylow p-subgroup — can be chosen equivariantly under any coprime group of automorphisms f

desk verdict Theorem B is a genuinely useful new counting tool, but the proof of the independent main theorem rests on an unjustified D=B reduction in Corollary 2.3. read the letter →

arxiv 2512.13406 v2 pith:MGMNJUGK submitted 2025-12-15 math.RT math.GR

Adele Maltempo , Carolina Vallejo This is my paper · ORCID

classification math.RTmath.GR MSC 20C1520C25

keywords McKayconjecturewithgroupautomorphismsp-solvablegroupsGlaubermancorrespondenceOkuyama-Wajimaargumenttheta-goodconjugacyclassesA-equivariantcharacterbijectionGallagher'scountingtheoremtriples

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 the McKay conjecture with group automorphisms for p-solvable groups, and in a relative form. If a group A of automorphisms acts on G with order coprime to |G| and fixes a Sylow p-subgroup P, then there is an A-equivariant bijection between the irreducible characters of degree prime to p of G and those of its local subgroup N_G(P); more generally, fixing a normal subgroup N and a P-invariant character θ of N, the bijection restricts to characters lying over θ on both sides. The engine is a generalized Gallagher count: |Irr^A(G|θ)| equals the number of θ-good conjugacy classes of G/N. This lets the authors follow the original Okuyama-Wajima argument for p-solvable groups, rather than invoking deeper classification results about endo-p-permutation modules. The counting result is stated with no assumption that the action be free, so it may be useful beyond McKay bijections.

What carries the argument

The argument is carried by two mechanisms. First, a Gallagher-type class-function count: with N, G normal in A and θ A-invariant, the vector space spanned by A-invariant irreducible characters of G over θ is isomorphic to the space of functions on a set of representatives of θ-good conjugacy classes of G/N; hence the character count is a class count. Here θ-good means every extension of θ to N⟨a⟩ is fixed by the centralizer of N a modulo N. Second, the Okuyama-Wajima transfer: for a group K with order not divisible by p and a p-subgroup P with KP normal, θ extends to KU exactly when the Glauberman correspondent θ* — the distinguished irreducible constituent of θ's restriction to the fixed-po

What would settle it

Compute the two sides of Corollary 2.3 in a small p-solvable example with C≤D=N_B(U)<B, for instance a semidirect product where U is a p-complement of H and its normalizer in B is smaller than B. If |Irr^{KD}(KS|θ)| and |Irr^D(S|θ*)| differ for some P-invariant θ with Glauberman correspondent θ*, the D=B reduction fails and Theorem A loses its support from this argument.

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Extended reading notes

Core claim

The central claim is Theorem A: for a finite group G with a Sylow p-subgroup P, a normal subgroup N, a P-invariant irreducible character θ of N, and a group A of automorphisms stabilizing both N and P, there exists an A-equivariant bijection between the irreducible characters of G of degree prime to p that lie over θ and the analogous characters of the subgroup N_G(P)N. To prove this, the paper establishes a generalized Gallagher count: whenever N and G are normal in A and θ is A-invariant, |Irr^A(G|θ)| equals the number of θ-good conjugacy classes of G/N, where an element is θ-good when all extensions of θ to the subgroups generated by N and that element are invariant under the relevant cen

Load-bearing premise

The proof of Corollary 2.3 says 'we can assume D=B' right after displaying the subgroup diagram, without showing how the general case C≤D≤B follows; since Theorem 2.4 later applies the corollary with D=N_B(U), a proper subgroup of B in general, the entire chain from Corollary 2.3 to Theorem A rests on this unstated reduction.

Editorial extensions

If this is right

For p-solvable groups, the automorphism-equivariant McKay bijection can now be proved without relying on the classification of endo-p-permutation modules; the argument needs only the Okuyama-Wajima extension transfer and the generalized character count.
The relative version above a fixed character θ means the McKay bijection is compatible with Clifford theory: it can be restricted to characters lying over any P-invariant irreducible character of a normal subgroup.
Theorem B gives a practical way to compute |Irr^A(G|θ)|: one only needs to inspect which conjugacy classes of G/N are θ-good, a group-theoretic condition.
Since the bijection is A-equivariant, any group of automorphisms that stabilizes P and N yields the same comparison; in particular, outer automorphisms acting on G cannot break the McKay bijection for p-solvable groups.
The generalized count applies to any A-invariant θ, not only in p-solvable or prime-to-p situations, so it is a general counting tool for invariant characters over a fixed character.

Reading between the lines

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

The counting theorem suggests a route to an automorphism-equivariant version of the Alperin-McKay conjecture for p-solvable blocks, where the relevant local structure is a maximal Brauer pair rather than a Sylow subgroup; the paper does not address blocks.
The proof's dependence on the abelian case of Okuyama-Wajima indicates that extending Theorem A beyond p-solvable groups would require a non-abelian version of the extension transfer; the paper notes such versions are known only through much heavier module-theoretic results.
One could test the limits of Theorem B by asking whether a similar 'good element' count holds for characters fixed by a Galois group, not just by a group of automorphisms; the paper mentions this as an open direction.
The unstated reduction D=B in Corollary 2.3 deserves a direct check: if the equality fails for a proper D=N_B(U), the proof's chain would need repair even if the theorem remains true.

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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

I'll skip the formalities. The genuinely new thing here is Theorem B: a Gallagher-type count of A-invariant characters above θ in terms of θ-good classes. The proof follows Navarro's construction and looks correct. It is a useful tool and could have legs beyond this paper — Corollary 1.4 is a nice bonus. I believe that part.

The problem is the route to the main theorem. Corollary 2.3 is the bridge: it supplies an Okuyama-Wajima style equality for subgroups D between C and B. Its proof says 'to simplify the notation, we can assume D=B' and never revisits the general case. This is not a notational convenience. In Theorem 2.4, the corollary is used with D = N_B(U), which is generally a proper subgroup of B and does not contain P. The D=B proof needs P inside the normalizer B, and the ambient group A = KB. For general D, you'd need to run the argument in the group KD with a different p-subgroup; the manuscript gives no such adaptation and no alternative reduction. Since Theorem 2.4 rests on Corollary 2.3 and Theorem 3.1 rests on Theorem 2.4, the independent proof of the headline theorem fails as written. The statement may well be true, and the gap might be patchable, but it's a load-bearing missing argument, not a typo.

The rest of the paper is honest and well-organized. The authors explicitly credit Rossi's theorem, so the novelty is the proof route, and that's exactly where the gap sits. The citation practices are fine. I'd also double-check the equivariance claims around the restriction bijections in Theorem 2.4, but that looks like a smaller matter.

So: read it for Theorem B and the strategy, but don't rely on the main theorem until the Corollary 2.3 reduction is fixed or replaced. If the authors supply that, it's a publishable independent proof; if not, it's a strong counting lemma with an unfinished application. I'd want a careful referee to see the fix before accepting the paper as is.

Referee Report

1 major / 4 minor

Summary. This paper proves Theorem A: if a finite group A acts on G stabilizing a normal subgroup N and a Sylow p-subgroup P, and G/N is p-solvable, then for every P-invariant θ∈Irr(N) there is an A-equivariant bijection Irr_{p'}(G|θ) → Irr_{p'}(N_G(P)N|θ). The proof is independent of Rossi's recent theorem and follows the classical Okuyama-Wajima strategy. The main ingredients are Theorem B, a new Gallagher-type count of A-invariant characters lying over θ in terms of θ-good conjugacy classes; Lemma 2.2, a reduction tool; Corollary 2.3, an equivariant version of the Okuyama-Wajima correspondence; Theorem 2.4, which compares p'-degree A-invariant characters of G and B over a product character; and a final induction in Theorem 3.1 along the lines of Navarro's proof of the p-solvable McKay conjecture.

Significance. If correct, the paper gives a self-contained proof of the automorphism-equivariant relative McKay conjecture for p-solvable groups, avoiding the deep classification results used in Rossi's approach (Ladisch, Dade, Turull, endo-p-permutation modules). Theorem B is a natural generalization of Gallagher's counting theorem and is likely to be of independent interest. The organization is clear and the arguments follow Navarro's book closely, making the proof checkable. The main issue identified below concerns the precise statement and proof of Corollary 2.3; it is local and fixable, and it does not appear to affect the final theorem.

major comments (1)

[Corollary 2.3] The proof of Corollary 2.3 declares "we can assume D=B" without the required justification. The reduction is valid only after replacing A by KD and B by D, using N_{KD}(P)=D because K∩B=C≤D. More importantly, the statement does not assume KS is normal in KD, yet the proof invokes Lemma 2.2, which requires G=KS to be a normal subgroup of the ambient group. This normality is not automatic from SŸD and KŸA; for example, in a semidirect product A=K⋊P with S a non-trivial subgroup of P, KS is not normalized by K. The application in Theorem 2.4 (S=U, D=N_B(U)) has KU normal in KD=N_A(KU) by construction, so the main result is not endangered; nevertheless, Corollary 2.3 must be restated with the missing normality hypothesis and the reduction to D=B must be derived explicitly.

minor comments (4)

[Corollary 2.3] When fixing the proof, please add the explicit observation N_{KD}(P)=D (every k∈K normalizing P lies in C) and state that the D=B case is applied to the ambient group KD.
[Theorem 2.4] The sentence "Notice that Z⊆P" should be justified: since Z is a normal p-subgroup of G, it is contained in every Sylow p-subgroup of G, in particular in P.
[Theorem 3.1] The letter A is reused for a set of orbit representatives in the proof of Theorem 3.1, and later B is reused for another set. This conflicts with the groups A and B; please use e.g. \mathcal{A} and \mathcal{B}.
[General] The header of the arXiv version contains typographical artifacts such as "McKa y" and "W ajima"; these should be cleaned up in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all load-bearing inputs are external standard results; the flagged Corollary 2.3 issue is a proof gap, not circularity.

full rationale

The paper's central claim, Theorem A, is derived from Theorem 2.4, which in turn is derived from Theorem 2.1 (the Okuyama–Wajima argument, cited externally from [OW80]) and Theorem B. Theorem B is a Gallagher-type counting theorem proved from Navarro's external Theorem 5.14 together with standard character-triple reductions; none of these inputs assumes the McKay conjecture with automorphisms or the target bijection. The cited sources [Nav18], [OW80], and [Isa76] are standard external works not authored by the present authors, and they are used with stated structural assumptions that do not include the target result. There are no fitted parameters presented as predictions, no self-citation chain used as a load-bearing justification, and no renaming of a known result as an independent derivation. The only concern raised by a close reading is that Corollary 2.3's proof says 'we can assume D=B' without deriving the reduction for proper subgroups D, although the later proof of Theorem 2.4 applies the corollary with D=N_B(U), which is in general a proper subgroup of B. That is a potential proof gap or correctness issue, not an instance of circularity: the proof of the D=B case does not invoke the general statement, the target theorem, or any fitted quantity. Consequently, no circular step is exhibited, and the circularity score is 0.

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

The central argument depends on standard but nontrivial character-theory machinery: Glauberman correspondence, Okuyama-Wajima, character triple isomorphisms, Schur-Zassenhaus. No free parameters or invented entities. The least secure entries are the extension of Okuyama-Wajima to the needed generality and the preservation of normalizer-fixed characters under restriction bijections.

assumptions (5)

domain assumption Okuyama-Wajima argument holds for an arbitrary p-subgroup P and abelian U/C, not only in the original stronger hypothesis.
The proof's engine. It is cited from [OW80] with the claim that stronger hypotheses are unnecessary, justified by [Nav18, Thm 5.10 and Cor 6.2]. If false in this generality, Theorems 2.4 and 3.1 collapse.
standard math Glauberman correspondence and its equivariance: B_θ = B_θ* and H_θ = H_θ* for the fixed-point subgroup C_K(P).
Used in Lemma 2.2, Corollary 2.3, and Theorem 2.4. Standard from [Nav18, Lem 2.1, Thm 2.9].
standard math Strong isomorphism of character triples preserves θ-goodness and p'-degree ratios.
Used in the central reduction in Theorem 3.1. Relies on [Nav18, Prob 5.4, Lem 5.15] and [Isa76, Def 11.23].
standard math The p-complement U/C of H/C from Schur-Zassenhaus exists, and B-conjugates of U are H-conjugates.
Used to build the subgroups U, KU, N_B(U), N_A(KU) in Theorem 2.4. Standard finite group theory.
domain assumption The restriction bijections from [Isa73, Lem 10.5]/[Nav18, Lem 6.8] preserve N_A(KU)- and N_B(U)-fixed characters.
Asserted without proof in Theorem 2.4. Plausible and standard in spirit, but load-bearing for the final count.

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Pith. "Pith review of The McKay conjecture with coprime group automorphisms and the Okuyama-Wajima argument." pith.science (2026). https://pith.science/paper/MGMNJUGK

@misc{pith2026251213406,
  author       = {Pith},
  title        = {Pith review of: The McKay conjecture with coprime group automorphisms and the Okuyama-Wajima argument},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MGMNJUGK}},
  note         = {Machine review of arXiv:2512.13406}
}

abstract

Let $A$ and $G$ be finite groups. Suppose that $A$ acts coprimely on $G$ stabilizing $N\triangleleft G$. Let $\theta \in \rm{Irr}(N)$ be $A$-invariant. We prove that the number of $A$-invariant irreducible characters of $G$ that lie over $\theta$ can be counted in terms of the $(A, \theta)$-good conjugacy classes of $G_\theta/N$, where $G_\theta$ is the inertia subgroup of $\theta$ in $G$. This result generalizes a classic result of Gallagher and can be used to prove the following: if $P$ is an $A$-invariant Sylow $p$-subgroup of $G$ and $G$ is $p$-solvable, then there exists an $A$-equivariant (McKay) bijection between the irreducible characters of degree prime to $p$ of $G$ and those of $\textbf{N}_G(P)$. While this is a consequence of a recent result of D. Rossi, our approach here is independent of Rossi's and follows the original idea of the proof of the McKay conjecture for $p$-solvable groups. In particular, we rely on the so-called Okuyama-Wajima argument to deal with characters above Glauberman correspondents.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Cabanes and B

M. Cabanes and B. Sp\"ath, The McKay conjecture on character degrees https://arxiv.org/abs/2410.20392, to appear in Ann. of Math

arXiv

[2] [2]

E. C. Dade, A correspondence of characters. In: The Santa Cruz Conference on Finite Groups (Santa Cruz, CA, 1979), Proc. Sympos. Pure Math. 37, Amer. Math. Soc., 401--403 (1980)

1979

[3] [3]

P. X. Gallagher, The number of conjugacy classes in a finite group. Math Z. 118 (1970), 175--179

1970

[4] [4]

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

1973

[5] [5]

I. M. Isaacs, Character theory of finite groups . Academic Press, New York-London, 1976. Pure and Applied Mathematics, No. 69

1976

[6] [6]

I. M. Isaacs, Characters of Solvable Groups . Graduate Studies in Mathematics 189 AMS, 2018

2018

[7] [7]

I. M. Isaacs, G. Malle and G. Navarro, A reduction theorem for the McKay conjecture, Invent. math. 170 (2007), 33--101

2007

[8] [8]

Ladisch, Character correspondences induced by magic representations

F. Ladisch, Character correspondences induced by magic representations. arXiv:1004.4538v2 (2011)

arXiv 2011