{"id":"876746d6-729e-4071-a8b5-2d50cd50205d","arxiv_id":"2512.13406","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"An automorphism-equivariant McKay bijection for p-solvable groups is proven by a generalized Gallagher character count and the Okuyama-Wajima argument.","lead":"The authors prove new counting formulas for characters fixed by group automorphisms and use them to give an independent, elementary proof of the McKay conjecture with automorphisms for p-solvable groups. The theorem was already known; the contribution is the proof route via the Okuyama-Wajima argument.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 2.3's proof assumes D=B without justifying the general C≤D≤B case, but Theorem 2.4 needs D=N_B(U), a proper subgroup of B.","rationale":"The reader's weakest assumption correctly identifies a real gap: Corollary 2.3 is needed in the manuscript for a proper subgroup D=N_B(U)≤B, but the proof only addresses D=B and gives no justification for the reduction. This is load-bearing because Theorem 2.4 depends on it, and Theorem 2.4 is the stepping stone to Theorem A's independent proof. I also examined whether the gap is easily repairable by adapting the D=B argument to general D, replacing A with KD and B with D. The adaptation seems plausible: Lemma 2.2's hypotheses can be checked (KD=(KS)D, S=KS∩D), and the Okuyama-Wajima theorem can still be invoked with the original ambient group A for subgroups U≤D≤B. However, the manuscript does not contain this adaptation, so as written the proof is incomplete. The verdict should remain CONDITIONAL pending clarification; the concern does not change the reader's assessment.","tokens_in":12701,"tokens_out":32180,"duration_ms":252305,"concrete_test":"Attempt a full proof of Corollary 2.3 for arbitrary C≤D≤B by modifying the D=B proof: set A'=KD, B'=D, replace every occurrence of A with A' and B with B', and verify that (i) Lemma 2.2 hypotheses hold (A'=(KS)D, H=KS∩D=S), and (ii) Theorem 2.1 can still be applied for U=⟨C,s,b⟩ with U≤D using the original ambient group A (which contains P and KP). If the adaptation goes through without new assumptions, the gap is expository; if it fails, produce a counterexample showing the equality |Irr_KD(KS|θ)| = |Irr_D(S|θ*)| can fail for D<B.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Corollary 2.3, the proof states 'To simplify the notation, we can assume that D=B' immediately after the subgroup diagram, but it does not derive how the general case C≤D≤B follows. The subsequent argument proves the equality only for D=B, where KD=A and the right-hand side is Irr_B(S|θ*). However, Theorem 2.4 later applies Corollary 2.3 with D=N_B(U), which is in general a proper subgroup of B and need not contain P. Thus the proof of Theorem 2.4 (and hence Theorem 3.1/Theorem A) relies on an unproved instance of Corollary 2.3. A direct adaptation of the D=B proof to general D would require replacing the ambient group A by KD and B by D, and checking that Lemma 2.2 still applies (A=GB, H=G∩B) and that the Okuyama-Wajima theorem can be used for subgroups U≤D. The manuscript does not supply this adaptation or any alternative reduction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12948,"tokens_out":29598,"duration_ms":274690,"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":[{"comment":"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.","section":"Corollary 2.3"}],"minor_comments":[{"comment":"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.","section":"Corollary 2.3"},{"comment":"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.","section":"Theorem 2.4"},{"comment":"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}.","section":"Theorem 3.1"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The gap in Corollary 2.3 is real but fixable by a short argument, and the application in Theorem 2.4 satisfies the missing hypothesis. I recommend requiring the authors to amend the statement and proof of Corollary 2.3 before publication. The rest of the paper appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nSo: 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.","headline":"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.","tokens_in":13434,"tokens_out":6041,"would_cite":true,"duration_ms":57034,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C15","20C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["McKay conjecture with group automorphisms","p-solvable groups","Glauberman correspondence","Okuyama-Wajima argument","theta-good conjugacy classes","A-equivariant character bijection","Gallagher's counting theorem","character triples"],"falsifier":"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.","tokens_in":12568,"feed_emoji":"🔢","tokens_out":8486,"duration_ms":78986,"temperature":0.7,"pith_summary":"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.","feed_headline":"Automorphism-invariant McKay bijection proved for p-solvable groups","feed_subtitle":"A generalized Gallagher count makes the proof elementary, bypassing deeper module classification.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Coprime automorphism McKay bijection: new elementary proof","A-equivariant McKay bijection via Okuyama-Wajima argument","Generalized Gallagher count settles McKay with automorphisms","Okuyama-Wajima: independent proof of automorphism McKay","McKay with automorphisms: elementary count via Gallagher"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Coprime automorphism McKay bijection: new elementary proof","A-equivariant McKay bijection via Okuyama-Wajima argument","Generalized Gallagher count settles McKay with automorphisms","Okuyama-Wajima: independent proof of automorphism McKay","McKay with automorphisms: elementary count via Gallagher"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1325,"prompt_tokens":796,"completion_tokens":529,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":444}},"tokens_in":540,"tokens_out":529,"duration_ms":5651,"temperature":1.0,"reasoning_tokens":444,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:27:52.590977+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}