{"id":"c51d2a7b-7a06-4aa5-8b03-7dc5228c9d41","arxiv_id":"2606.29295","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every ideal of coprime order in a finite skew brace admits a complement that is a sub-skew brace.","lead":"Finite skew braces admit complements for ideals of coprime order, exactly as groups do under Schur–Zassenhaus. The result lets researchers split brace extensions without extra solubility or cohomological hypotheses.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates Feit–Thompson as the only non-elementary ingredient and notes that it is used solely to obtain conjugacy inside the D_π property, not bare existence. Because the paper claims only existence of a complement, the dependence is harmless: ordinary Schur–Zassenhaus already supplies a Hall π-subgroup of each factor, and the trifactorised theorem then assembles them into a common complement. The line-by-line argument that H_K = H_C = H_D is a genuine sub-skew brace of order |B/I| is self-contained and does not rely on further solubility assumptions. Consequently the reader’s ACCEPT / high-confidence assessment stands; no adjustment is required.","tokens_in":7368,"tokens_out":540,"duration_ms":4479,"concrete_test":"Independently verify that the three sets H_K, H_C, H_D defined after the application of Theorem 2.3 coincide by recomputing the product (h,h)=(k,1)(0,c) inside G_π and checking that |H_K|=|H_C|=|H_D|=b forces equality; if they fail to coincide for a concrete coprime ideal (e.g., a trivial brace of order 6 with ideal of order 2), the extraction of the complement would be incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem A) is that a coprime ideal I of a finite skew brace B admits a complement sub-skew brace H. The proof reduces the problem to the classical Schur–Zassenhaus theorem and the Hall theorem for trifactorised groups (Theorem 2.3 of [1]) by embedding B into the trifactorised holomorph G = (B,+) ⋊_λ (B,·). Lemma 2.2 correctly extracts the D_π property from Feit–Thompson + classical SZ for each of G, C and D; the subsequent extraction of a common Hall π-set H that is simultaneously a subgroup of both operations is explicit and order-theoretic. The only external dependence is Feit–Thompson (already classical) and the cited trifactorised Hall theorem; neither introduces a hidden hypothesis that would invalidate existence of the complement. The conjugacy question is deliberately left open, and the counter-example correctly shows that the stronger containment property fails. No internal inconsistency or missing step appears.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper proves an analogue of the classical Schur–Zassenhaus theorem for finite skew braces: if B is a finite skew brace and I is an ideal with |I| and |B/I| coprime, then I admits a complement (a sub-skew brace H satisfying I ∩ H = {0}, (B,+) = I + H and (B,·) = IH). The argument embeds B into the associated trifactorised holomorph G = (B,+) ⋊_λ (B,·), verifies that G, C ≅ (B,·) and D (the diagonal) all satisfy the D_π-property via a short lemma that combines classical Schur–Zassenhaus with Feit–Thompson, and then invokes the Hall theorem for trifactorised groups from [1] to extract a common Hall π-subgroup that is simultaneously a subgroup of both operations. An explicit counter-example shows that the stronger containment property of Sylow theory fails in general for skew braces. Conjugacy of complements is deliberately left open.","tokens_in":7566,"tokens_out":719,"duration_ms":6594,"significance":"The result fills a natural gap in the emerging Sylow/Hall theory of finite skew braces. Earlier work established existence of Sylow and (under solubility) Hall sub-skew braces; the present note shows that the coprime-complement statement holds without extra hypotheses, by a clean reduction to classical group theory and the trifactorised-group machinery of [1]. The proof is short, fully written out, and free of ad-hoc parameters or circular appeals. The counter-example clarifying the limits of containment is a useful addition. The dependence on Feit–Thompson is classical and correctly scoped; the paper therefore supplies a solid, reusable tool for the study of extensions and Hopf–Galois structures on skew braces.","major_comments":[],"minor_comments":[{"comment":"In the proof of Theorem A the three sets HK, HC, HD are shown to coincide by a short order argument after the inclusion HD ⊆ HK ∩ HC; a single sentence noting that the same conclusion follows from the three factorisations of Gπ would make the identification even more transparent.","section":null},{"comment":"Lemma 2.2 is used three times in essentially identical fashion; a brief remark that the same argument applies verbatim to any group possessing a normal Hall π′-subgroup would avoid the slight repetition.","section":null},{"comment":"The arXiv identifiers of the very recent preprints [1], [3], [4], [10] and [11] will need updating once they appear in print; the present citations are otherwise accurate.","section":null},{"comment":"Page 1, line 3 of the abstract: the phrase “admits a complement in B” is clear from context, but a parenthetical reminder of the precise meaning (sub-skew brace H with the three listed properties) would help readers who skip the introduction.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, self-contained and relies on a preprint ([1]) that is still under review; if that companion paper is delayed, the journal may wish to ask the authors for a short self-contained appendix extracting the precise statement of the trifactorised Hall theorem that is used. Otherwise the paper is ready for publication as a note."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This note gives a genuine Schur–Zassenhaus theorem for finite skew braces: a coprime ideal always has a complement that is a sub-skew brace for both operations. Earlier complement results needed extra cohomological or structural hypotheses; Damele removes them by embedding the brace into the usual trifactorised holomorph and applying the Hall theorem for trifactorised groups from Ballester-Bolinches et al.\n\nWhat works well is the reduction. Lemma 2.2 correctly extracts the D_π property from classical Schur–Zassenhaus plus Feit–Thompson for the three factors G, C and D. The extraction of the common Hall set H that is simultaneously additive and multiplicative is explicit and order-theoretic; once you have |H| = |B/I| and the coprimeness, the complement relations fall out immediately. The counter-example showing that the stronger containment property fails is concrete and useful. Citations are appropriate and the argument is short enough to check line-by-line.\n\nSoft spots are minor. The conjugacy half is left open, which the author flags honestly; that is a real open question rather than a gap in the existence proof. The reliance on Feit–Thompson is classical and correctly used only to get conjugacy inside the ambient groups so that the D_π property holds; existence of Hall subgroups would survive without it, but the cited trifactorised theorem needs the full D_π package. No circularity, no free parameters, no hidden hypotheses that break the claim.\n\nThis is for people working on braces, Hopf–Galois structures or set-theoretic Yang–Baxter solutions who need clean splitting results. It is a solid incremental advance, not a field-reorganizing one. I would send it to a serious referee without hesitation; the math is transparent and the result is usable.","headline":"Clean, hypothesis-free Schur–Zassenhaus for finite skew braces via trifactorised groups; short and solid.","tokens_in":8143,"tokens_out":464,"would_cite":true,"duration_ms":4047,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T25","20D20"],"pacs":[],"model":"grok-4.5","headline":"Finite skew braces admit complements for every ideal of coprime order and index.","keywords":["skew braces","Schur–Zassenhaus theorem","Hall subgroups","trifactorised groups","complements","ideals"],"falsifier":"Exhibit a finite skew brace B and an ideal I with gcd(|I|,|B/I|)=1 for which no sub-skew brace H satisfies both I ∩ H = {0} and |H| = |B/I|; equivalently, show that the three projected Hall subgroups of the associated trifactorised group never coincide.","tokens_in":8251,"feed_emoji":"➗","tokens_out":898,"duration_ms":6686,"temperature":0.7,"pith_summary":"Skew braces are algebraic structures that carry two group operations linked by a simple compatibility rule; they arose as a way to organise set-theoretic solutions of the Yang–Baxter equation. This paper proves that the classical Schur–Zassenhaus theorem survives in this larger setting: whenever an ideal I of a finite skew brace B has order coprime to the order of the quotient B/I, there exists a complementary sub-skew brace H that meets I only at the identity and multiplies with I to recover both of B’s group structures. The argument works by embedding B into a carefully chosen trifactorised group, applying the ordinary Schur–Zassenhaus theorem together with a Hall theorem for trifactorised groups, and then reading the resulting Hall subgroup back as the desired complement. A short counter-example shows that the stronger containment property familiar from Sylow theory does not hold in general.","feed_headline":"Coprime ideals in finite skew braces always split","feed_subtitle":"A classical complement theorem survives the jump from groups to two-operation braces","key_machinery":"The associated trifactorised group G = (B,+) ⋊_λ (B,·) = KC = KD = DC, together with the D_π-property of its three factors; a Hall π-subgroup of G that is itself trifactorised yields, under the natural projections, a common sub-skew brace of the required order that complements I.","core_discovery":"If B is a finite skew brace and I is an ideal whose order is coprime to the order of the quotient B/I, then I possesses a complement: a sub-skew brace H such that I ∩ H is trivial, the additive group of B is the sum I + H, and the multiplicative group of B is the product IH.","pith_inferences":["The missing conjugacy statement for complements is likely the next natural target; a counter-example or a positive result under solubility assumptions would clarify how much of classical Schur–Zassenhaus really survives.","The same embedding technique may produce analogues of other classical complement theorems (for example Gaschütz’s theorem) once suitable D_π-type hypotheses are verified for the trifactorised group.","Because the construction is entirely group-theoretic, it should specialise cleanly to ordinary braces and to left-nilpotent skew braces, possibly recovering earlier partial results with shorter proofs."],"forward_implications":["Every ideal of coprime order and index in a finite skew brace is a direct factor with respect to both group operations.","Existence of Hall π-sub-skew braces follows at once whenever the complementary order is a π-number.","Complement problems for skew braces no longer require extra cohomological or nilpotency hypotheses when the orders are coprime.","The same trifactorised-group method can be reused to transfer other Hall-type results from groups to skew braces."],"fun_headline_variants":["Schur-Zassenhaus theorem extends to finite skew braces","Coprime ideals of finite skew braces admit complements","Finite skew braces split over ideals of coprime order","Complements exist for coprime ideals in skew braces","Skew braces inherit Schur-Zassenhaus when orders are coprime"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The proof relies on the theorem that every group of odd order is soluble, so that at least one of the two coprime factors is soluble and the conjugacy half of the classical Schur–Zassenhaus theorem can be applied inside the trifactorised group.","fun_headline_variants_meta":{"raw":{"variants":["Schur-Zassenhaus theorem extends to finite skew braces","Coprime ideals of finite skew braces admit complements","Finite skew braces split over ideals of coprime order","Complements exist for coprime ideals in skew braces","Skew braces inherit Schur-Zassenhaus when orders are coprime"]},"model":"grok-4.5","effort":"low","cost_usd":0.004382,"raw_usage":{"total_tokens":1183,"prompt_tokens":583,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":43820000,"prompt_tokens_details":{"text_tokens":583,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":513,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":583,"tokens_out":87,"duration_ms":5077,"temperature":1.0,"reasoning_tokens":513,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T17:03:43.464377+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a finite skew brace B and an ideal I with gcd(|I|,|B/I|)=1 for which no sub-skew brace H satisfies both I ∩ H = {0} and |H| = |B/I|; equivalently, show that the three projected Hall subgroups of the associated trifactorised group never coincide.","supporting_citations":[],"review_version":2}