{"id":"6e62da55-7a11-435b-bb56-bdd31e426efc","arxiv_id":"2607.14059","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A finite skew brace with additive group PSU_5(64)×A_5 and multiplicative group Aut(PSU_5(64)) is constructed, affirmatively solving Kourovka Problem 20.109.","lead":"This paper constructs a finite skew brace—an algebraic structure used to study the Yang–Baxter equation—whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. The example affirmatively answers an open problem from the Kourovka Notebook and shows how different the two sides of a skew brace can be.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The auxiliary brace C depends on the triviality of an action α recalled from Tsang; if α is nontrivial, the direct-product shape (C,∘)≅F20×C3 fails and Theorem 1.1 collapses.","rationale":"Proposition 2.1 is the only ingredient not fully proved in the paper; it imports a construction from Tsang. The α-triviality step is the gap within that recollection. If α were nontrivial, the multiplicative group of C would not be the required direct product F20×C3, and the homomorphism θ used to build B would not exist. The rest of the argument (Lemma 2.2, Proposition 3.1, and the final group-theoretic checks) is straightforward and internally consistent. The reader correctly identified this as the weakest assumption. I do not see a demonstrated flaw, but the step is sufficiently opaque that a computational/independent re-derivation is warranted. The paper's own logic is coherent, so the verdict remains ACCEPT (moderate confidence); a failed check would, however, be fatal. Hence UNCHANGED.","tokens_in":3801,"tokens_out":32483,"duration_ms":291350,"concrete_test":"In GAP, reconstruct Tsang's q=5 construction: set L=PGL(2,5), N=PSL(2,5), choose a Singer cycle X and point stabilizer Y with L=XY. Following the construction in [7, Prop. 2.7], build the regular subgroup Γ≤Hol(N) whose multiplicative group is X0⋊_α Y. Compute the action of a generator of Y/Y0 on X0 via the identification with X/X0; verify it is the identity (equivalently, that the element of order 2 in X commutes with X0). Then compute the isomorphism type of Γ and confirm it is F20×C3. If α is nontrivial or Γ is not F20×C3, Theorem 1.1's construction fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the claim in Proposition 2.1 that the action α of Y on X0 is trivial, justified by 'Since X≃C6 is abelian, this action is trivial' (§2.1). This is correct only if α is the abstract action obtained from the identification Y/Y0≅X/X0 and is given by conjugation by an element of X. It cannot be the ordinary conjugation action of Y on X0 inside L=PGL2(5): if Y normalized X0, then X0Y would be a subgroup of order 60, hence A5, contradicting that Y contains odd permutations. So α is a non-obvious construction-specific action. If α were the nontrivial automorphism of C3 (inversion), the multiplicative group of C would be C3⋊F20 rather than C3×F20; then the isomorphism θ:(C,∘)→H in Theorem 1.1, which needs the C3 factor central, would be unavailable. The paper gives no independent verification of this recalled action, so the existence of the auxiliary brace C is the least secure pillar.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a finite left skew brace B=(B,+,∘) with additive group PSU5(64)×A5 and multiplicative group Aut(PSU5(64)). This gives a positive answer to Kourovka Problem 20.109, which asks for a finite skew brace with perfect additive group and non-perfect almost simple multiplicative group. The construction combines: (i) an auxiliary skew brace C with additive group A5 and multiplicative group F20×C3, obtained as a specialization of a construction of Tsang; (ii) the splitting of Aut(PSU5(64)) over PSU5(64), so that Aut(PSU5(64))=PSU5(64)⋊H with H≅F20×C3; and (iii) a semidirect product of the trivial brace on PSU5(64) with C via an isomorphism θ:(C,∘)→H. The verification of the brace axioms is delegated to Lemma 2.2, and the perfect/non-perfect assertions are straightforward.","tokens_in":4049,"tokens_out":27557,"duration_ms":226285,"significance":"If correct, the paper solves an open problem from the Kourovka Notebook. The construction is elegant and uses a clean semidirect product of braces, with all group-theoretic inputs taken from published sources. The main strength is the explicit identification of the additive and multiplicative groups, and the care in noting that the example is necessarily one-sided. The authors correctly rely on the splitting theorem for Aut(PSU5(64)) and on Tsang's auxiliary brace; these are external but established results. The paper is concise and checkable.","major_comments":[],"minor_comments":[{"comment":"The definition of the action α is very terse. The phrase 'induced by conjugation by the complement of X0 in X' is load-bearing, and the reader may initially confuse it with the ordinary conjugation action of Y on X0 inside L, which is not well-defined because Y does not normalize X0. Since X is abelian, the stated triviality is correct for the intended action, but a brief explicit description (e.g., via the isomorphism Y/Y0 ≅ X/X0) would remove ambiguity. This is a presentation issue, not a technical gap.","section":"§2.1, Proposition 2.1"},{"comment":"Typographical error: 'moltiplicative' should be 'multiplicative' in the sentence '... can occur as the moltiplicative group of a skew brace ...'.","section":"§1, Introduction"},{"comment":"Minor formatting issue: 'Theorem1.1' is missing a space in the abstract.","section":"Abstract"},{"comment":"The splitting criterion cited from [5] is stated in a condensed form. A more precise pointer to the exact statement in [5] (including the list of exceptions) would help the reader verify the applicability to PSU5(64).","section":"§3, Proposition 3.1"}],"recommendation":"accept","confidential_remarks":"The paper is sound and the main construction is clear. The only point that might deserve an editorial request is a fuller explanation of the action α in Proposition 2.1; however, since the result is imported from Tsang [7] and the proof sketch is essentially a recollection, I do not consider this blocking. The stress-test concern about the triviality of α does not land as stated: the proof does not claim that Y normalizes X0, and the action is defined via conjugacy by a complement in X, which is abelian."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper settles an explicit open problem from the Kourovka Notebook: it constructs a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. The construction is new, even though the ingredients are known. The authors take Tsang's order-60 brace and the known splitting of Aut(PSU5(64)), then assemble them with a semidirect product. That assembly is clean and correct.\n\nWhat the paper does well: Lemma 2.2 is a straightforward semidirect product construction for skew braces, with a correct lambda map. Proposition 3.1's outer automorphism computation checks out: for PSU5(64), Out is F20×C3, and the Lucchini–Menegazzo–Morigi splitting criterion gives gcd(13,5,6)=1, so Aut splits. The final perfectness argument is standard and correct. The paper also correctly notes it is only a left skew brace, contrasting with the two-sided theorem where perfect additive forces perfect multiplicative.\n\nThe soft spot is Proposition 2.1. The auxiliary brace is imported from Tsang, and the one-line justification that the action α is trivial because X is cyclic is terse. The stress-test worry about ordinary conjugation inside PGL2(5) is a misreading: the paper explicitly says α factors through Y/Y0 and is induced by conjugation from the complement of X0 in X. Since X is abelian, that action is indeed trivial on X0. So the claim is correct, but the exposition could be clearer. I would ask the authors to expand that sentence in the published version, not because the reasoning is wrong but because it is too compressed for a reader who hasn't internalized the Tsang construction.\n\nThere is no circularity, no fitted parameters, and the external results are cited precisely. The paper is a solid, specialized existence proof. It does not develop new theory, but it answers a recognized open problem and gives a concrete example that may be useful for testing conjectures in skew brace theory.\n\nAudience: skew brace researchers and anyone tracking Kourovka problems. It deserves a serious referee; the referee should verify the two imported results, but the paper's own algebra is sound.\n\nRecommendation: send to peer review, with minor revision requested for the clarification in Proposition 2.1.","headline":"This paper answers Kourovka Problem 20.109 by constructing a finite skew brace with perfect additive group PSU5(64)×A5 and non-perfect almost simple multiplicative group Aut(PSU5(64)); the proof is sound, with the only real soft spot being a terse imported action.","tokens_in":4535,"tokens_out":3830,"would_cite":true,"duration_ms":63240,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16T25","20D05","20D06","20E32"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a finite skew brace whose additive group is perfect and whose multiplicative group is almost simple but non-perfect, giving an affirmative answer to a listed open problem.","keywords":["skew brace","perfect group","almost simple group","projective unitary group","automorphism group","semidirect product","Kourovka Notebook","Yang-Baxter equation"],"falsifier":"Explicitly compute the action α: Y → Aut(X_0) in the q=5 construction (where X_0 ≃ C_3 and Y ≃ F_20); if some element of Y acts non-trivially on X_0, then the auxiliary skew brace's multiplicative group is not F_20 × C_3, and the isomorphism θ used to build the semidirect product cannot be defined.","tokens_in":3675,"feed_emoji":"🧮","tokens_out":6547,"duration_ms":51079,"temperature":0.7,"pith_summary":"The paper constructs a finite skew brace B with additive group PSU_5(64) × A_5 and multiplicative group Aut(PSU_5(64)). This is the first known example where the additive group is perfect while the multiplicative group is almost simple yet not perfect, resolving an open problem from the Kourovka Notebook. The construction combines a small skew brace whose additive group is A_5 with the automorphism group of the projective unitary group PSU_5(64), using a semidirect product of skew braces. The result matters because it shows that, for left skew braces, perfection of the additive group imposes no such perfection on the multiplicative side.","feed_headline":"Skew brace pairs perfect additive with almost simple multiplicative","feed_subtitle":"The construction answers open problem 20.109 with groups PSU5(64) times A5 and Aut(PSU5(64)).","key_machinery":"The key machinery is the semidirect product of skew braces: if C is a skew brace and θ: (C,∘) → Aut(S,+) is a homomorphism, then S × C becomes a skew brace whose multiplicative group is the semidirect product (S,+) ⋊_θ (C,∘). The paper applies this with S = PSU_5(64) (as an additive group) and with C the auxiliary skew brace from an existing construction, whose additive group is A_5 and whose multiplicative group is F_20 × C_3. The argument also depends on the splitting of the exact sequence 0 → Inn(PSU_5(64)) → Aut(PSU_5(64)) → Out(PSU_5(64)) → 0, which provides a complement H ≃ F_20 × C_3 inside Aut(PSU_5(64)). The lambda map of the resulting brace is λ_{(s,x)}(t,y) = (θ_x(t), λ^C_x(y)), a","core_discovery":"The central claim is that there exists a finite skew brace B = (B,+,∘) with (B,+) ≃ PSU_5(64) × A_5 and (B,∘) ≃ Aut(PSU_5(64)). Since A_5 and PSU_5(64) are non-abelian simple groups, the additive group is perfect, while the multiplicative group, being isomorphic to Aut(PSU_5(64)), is almost simple with socle PSU_5(64) but not perfect because it surjects onto a solvable outer automorphism group F_20 × C_3. The proof builds B as a semidirect product Triv(PSU_5(64)) ⋊_θ C, where C is an auxiliary skew brace with additive group A_5 and multiplicative group F_20 × C_3, and θ is an isomorphism from (C,∘) to a complement of the inner automorphism group in Aut(PSU_5(64)); the existence of such a com","pith_inferences":["The same template may work for other finite simple groups S whose automorphism extension splits and whose outer automorphism group is the multiplicative group of a skew brace with perfect additive group; the splitting criterion used here could be scanned for further examples.","If the auxiliary action in the q=5 construction were non-trivial, the whole example would fail; this suggests that the answer to the problem is underwritten by a delicate coincidence that may not survive in related families.","The semidirect construction could be iterated: starting from this B and another skew brace with suitable additive group, one might build braces with even richer perfect/almost-simple combinations.","One could test whether the associated set-theoretic solution of the Yang–Baxter equation is bijective or involutive, which might connect this group-theoretic example to wider questions about the classification of such solutions."],"forward_implications":["Kourovka Problem 20.109 has a positive answer: a finite skew brace with perfect additive group and non-perfect almost simple multiplicative group exists.","The example is necessarily a left skew brace and not two-sided, because a finite two-sided skew brace with perfect additive group would have perfect multiplicative group.","The multiplicative group Aut(PSU_5(64)) is almost simple with socle PSU_5(64), so the additive group of the brace is exactly the direct product of the socle and A_5.","The construction yields a new family of skew braces built from automorphism groups of finite simple groups via a semidirect product, broadening the known supply of non-trivial Yang–Baxter solutions.","Since the additive group is a direct product of two non-abelian simple groups, the brace gives a concrete example where the additive structure is a perfect group with no abelian quotients while the multiplicative structure has a non-trivial solvable quotient."],"fun_headline_variants":["Skew brace pairs perfect additive with almost simple multiplicative","Answer to Kourovka 20.109 via a finite skew brace","PSU5(64)×A5 as additive, Aut(PSU5(64)) as multiplicative","Finite skew brace with perfect additive, non-perfect almost simple multiplicative","Construction pairs PSU5(64)×A5 with Aut(PSU5(64))"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction depends on the assertion that in the auxiliary skew brace with additive group A_5, the conjugation action of the multiplicative stabilizer on the subgroup C_3 is trivial; if that action were non-trivial, the multiplicative group would not be the direct product F_20 × C_3 and the isomorphism θ would not exist.","fun_headline_variants_meta":{"raw":{"variants":["Skew brace pairs perfect additive with almost simple multiplicative","Answer to Kourovka 20.109 via a finite skew brace","PSU5(64)×A5 as additive, Aut(PSU5(64)) as multiplicative","Finite skew brace with perfect additive, non-perfect almost simple multiplicative","Construction pairs PSU5(64)×A5 with Aut(PSU5(64))"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1151,"prompt_tokens":740,"completion_tokens":411,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":307}},"tokens_in":484,"tokens_out":411,"duration_ms":4687,"temperature":1.0,"reasoning_tokens":307,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T02:55:12.667628+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Explicitly compute the action α: Y → Aut(X_0) in the q=5 construction (where X_0 ≃ C_3 and Y ≃ F_20); if some element of Y acts non-trivially on X_0, then the auxiliary skew brace's multiplicative group is not F_20 × C_3, and the isomorphism θ used to build the semidirect product cannot be defined.","supporting_citations":[],"review_version":1}