{"id":"88ad0418-351f-4992-8031-08b2ea1d7bef","arxiv_id":"2411.15968","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In finite q-solvable groups, the p′-degree irreducible characters are all q′-degree exactly when some Sylow p-subgroup lies inside the normalizer of a Sylow q-subgroup and the derived subgroup of that Sylow q-subgroup is centralized by P.","lead":"The paper proves a precise criterion, involving how Sylow subgroups are normalized, for when every irreducible character whose degree avoids one prime also avoids another prime in finite q-solvable groups. It confirms a speculation from 1998 and extends earlier theorems to a wider class of groups using the recently proved McKay conjecture and the classification of finite simple groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The internal reductions are coherent, but the central theorems inherit load-bearing risk from the unrefereed preprints [CS24] and [LNPT24], so the verdict should be conditional.","rationale":"The reader's weakest_assumption correctly identifies [CS24] as the load-bearing external input. My independent reading of the proof confirms the internal logic is sound: the reduction in Proposition 2 via the degree-divisibility bijection and the McKay equality is valid assuming [CS24]; the Fong-Swan argument in Theorem A correctly lifts q-Brauer characters; Theorem 4's induction steps are terse but reconstructible, and Theorem 3 from [LNPT24] supplies the final abelian-Hall-subgroup conclusion. The only serious risk is that the main theorems are conditional on two very recent, not-yet-refereed deep results, one of which ([CS24]) is explicitly acknowledged by the author. Since the preprint itself presents these as theorems but the external verdicts are not settled, the appropriate editorial decision is a conditional acceptance pending verification of [CS24] and [LNPT24]. This matches the reader's caveat and does not allege any error in the paper's own reductions.","tokens_in":7078,"tokens_out":19974,"duration_ms":184800,"concrete_test":"Inspect the statement and proof of the main theorem in arXiv:2410.20392 [CS24] to confirm that the equality |Irr_{p'}(G)| = |Irr_{p'}(N_G(P))| is proved for all finite groups, with no hidden p-solvability hypothesis. If the equality is restricted, re-run Proposition 2 with the restricted statement and see whether the proof of Theorem A still goes through. For Theorem B, confirm that [LNPT24, Theorem A] has been refereed and that its 'does not involve A7' hypothesis is satisfied by every q-solvable group for q in {2,3,5,7}; if not, the final step of Theorem 4 loses its justification.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The weakest point is external. Proposition 2 applies [CS24] to assert |Irr_{p'}(G)| = |Irr_{p'}(N_G(P))| for a q-solvable group G that need not be p-solvable. If [CS24] is incomplete or inapplicable outside p-solvable groups, the iff in Theorem A collapses; the paper's own introduction says Theorem A depends on the McKay conjecture being true, and [CS24] (arXiv:2410.20392) is not yet refereed. Theorem B depends on [LNPT24, Theorem A] (quoted as Theorem 3), a submitted preprint that relies on classification and on the inductive McKay conditions. I checked the internal reductions: Proposition 2's cardinality argument, the Fong-Swan lifting in Theorem A, and the induction steps in Theorem 4 are coherent and I found no local mathematical gap. The concern is not a claimed error in this paper but the fact that its two main theorems are only as secure as two unrefereed deep preprints.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":7149,"tokens_out":21942,"duration_ms":183362,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Section 2, proof of Theorem A"},{"comment":"There are two occurrences of the typo 'Propostion' where 'Proposition' is intended.","section":"Section 3, proof of Theorem B"},{"comment":"The phrase 'arguing as before by for the prime q' contains a typo ('by for' should be 'for').","section":"Section 3, proof of Theorem B, Step 1"},{"comment":"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.","section":"Section 2, Proposition 2"}],"recommendation":"minor_revision","confidential_remarks":"The paper's two main theorems depend essentially on two unrefereed preprints: [CS24] (the McKay conjecture) and [LNPT24]. The author acknowledges this dependence. I did not find internal errors, but the correctness of the theorems is contingent on these external results. I recommend that the editor verify the status of [CS24] and [LNPT24] (for example, whether they have been accepted or certified by other experts). If they are correct, the paper is suitable for publication after minor editorial fixes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":7812,"tokens_out":2832,"would_cite":true,"duration_ms":24652,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20C15","20C20","20D20","20D10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $C_{Q'}(P)=1$.","keywords":["irreducible characters","character degrees","q-solvable groups","Sylow normalizers","q-blocks","defect groups","McKay conjecture","Glauberman correspondence"],"falsifier":"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.","tokens_in":92,"feed_emoji":"🎯","tokens_out":12042,"duration_ms":167076,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two local subgroup checks characterize p'-degree character sets","feed_subtitle":"For q-solvable groups, the just-proved McKay conjecture removes a long-standing separability condition.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Proves the McKay conjecture, supplying the cardinality bijection between $p'$-degree characters used in Proposition 2.","marker":"[CS24]"},{"why":"Proves the degree-divisibility McKay bijection for solvable groups, the base case of Theorem 1.","marker":"[Tur07]"},{"why":"Extends the degree-divisibility McKay bijection to $q$-solvable groups given Glauberman degree divisibility.","marker":"[Riz19]"},{"why":"Proves the Glauberman degree-divisibility needed to complete Theorem 1 after a reduction to simple groups.","marker":"[Gec20]"},{"why":"Reduces Glauberman degree divisibility to simple groups, an input used by Geck's proof.","marker":"[HT94]"},{"why":"Proves the Brauer-character version for $q$-solvable groups used to pass from Fong–Swan to normalizer containment in Theorem A.","marker":"[BNRS22]"},{"why":"Original theorem for $\\{p,q\\}$-separable groups that Theorem A extends, and source of the $\\mathrm{PSL}(2,3^5)$ example showing $p$-solvability is insufficient.","marker":"[NW98]"},{"why":"Original block theorem for $\\{p,q\\}$-separable groups that Theorem B extends, including the $J_1$ principal-block counterexample without a separability condition.","marker":"[NW01]"},{"why":"Supplies the classification-dependent relative $\\pi'$-degree theorem used as Theorem 3 in the proof of Theorem B.","marker":"[LNPT24]"},{"why":"Provides the block-induction reduction with $(B,Q)$-good characters used in the proof of Theorem B.","marker":"[MR23]"}],"fun_headline_variants":["p-solvability dropped: new character degree local subgroup theorem","Character degrees and local subgroups: beyond p-solvable groups","McKay step: no p-solvability needed for Navarro-Wolf results","Local subgroup condition characterizes degree sets without p-solvability","New proof removes p-solvability from Navarro-Wolf character theorems"],"cache_read_input_tokens":9856,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["p-solvability dropped: new character degree local subgroup theorem","Character degrees and local subgroups: beyond p-solvable groups","McKay step: no p-solvability needed for Navarro-Wolf results","Local subgroup condition characterizes degree sets without p-solvability","New proof removes p-solvability from Navarro-Wolf character theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000462,"raw_usage":{"total_tokens":2306,"prompt_tokens":938,"completion_tokens":1368,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":554,"completion_tokens_details":{"reasoning_tokens":1275}},"tokens_in":554,"tokens_out":1368,"duration_ms":10041,"temperature":1.0,"reasoning_tokens":1275,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:43:53.796504+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}