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A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.14059 v1 pith:PBQLIOYG submitted 2026-07-15 math.GR

Massimiliano Di Matteo , Maria Ferrara This is my paper · ORCID
classification math.GR MSC 16T2520D0520D0620E32
keywords skewbraceperfectgroupalmostsimpleprojectiveunitaryautomorphismsemidirectproductKourovkaNotebookYang-Baxterequation
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

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

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.

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.

minor comments (4)
  1. [§2.1, Proposition 2.1] 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.
  2. [§1, Introduction] Typographical error: 'moltiplicative' should be 'multiplicative' in the sentence '... can occur as the moltiplicative group of a skew brace ...'.
  3. [Abstract] Minor formatting issue: 'Theorem1.1' is missing a space in the abstract.
  4. [§3, Proposition 3.1] 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).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction relies on external, independently published inputs and does not assume its conclusion.

full rationale

The paper's derivation chain is: (1) import an auxiliary skew brace C with additive group A5 and multiplicative group F20×C3 from Tsang [7]; (2) prove a semidirect product construction for skew braces (Lemma 2.2); (3) invoke the Lucchini–Menegazzo–Morigi splitting criterion and standard facts on Out(PSU5(64)) to obtain Aut(PSU5(64)) ≅ S ⋊ H with H ≅ F20×C3; (4) combine these via the semidirect product to define B. None of these steps assumes Theorem 1.1, and none fits a parameter to the target conclusion. The auxiliary brace and the splitting criterion come from independent published sources, not from self-citations or from a uniqueness theorem imported from the present authors. The potentially delicate assertion in Proposition 2.1 that the action α is trivial because X ≅ C6 is abelian is a group-theoretic verification inside a cited external construction; even if its correctness were questioned, that would be a mathematical gap or risk, not circularity, because the cited result does not contain the conclusion of Theorem 1.1 and is not authored by the present paper's authors. The final skew brace is explicitly constructed, and the equalities (B,+) ≅ S×A5 and (B,∘) ≅ Aut(S) follow by construction from the external inputs. The paper is therefore externally benchmarked and self-contained against published results, with no circular dependence on its own claim.

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

The paper introduces no free parameters and no new postulated entities. The construction is a composition of three imported results—Tsang's order-60 brace, the standard outer-automorphism-group formula, and the Lucchini–Menegazzo–Morigi splitting criterion—plus an explicitly proved semidirect-product lemma. The load-bearing imported items are listed as axioms.

assumptions (5)
  • domain assumption Out(PSU_n(p^m)) = ⟨δ⟩⋊⟨φ⟩ with |δ|=gcd(n, p^m + 1), |φ|=2m, δ^φ = δ^p
    Used in Prop 3.1 to compute Out(PSU_5(64)) ≅ (C_5⋊C_4)×C_3; cited from Costantini–Lucchini–Nemmi [1].
  • domain assumption Aut(PSU_5(64)) splits over Inn(PSU_5(64)), because gcd((q+1)/d, d, m) = gcd(13,5,6) = 1
    Used in Prop 3.1 to get a complement H ≅ Out(PSU_5(64)) inside Aut(PSU_5(64)); criterion from Lucchini–Menegazzo–Morigi [5].
  • domain assumption There exists a skew brace C with additive group A_5 and multiplicative group (C_5⋊C_4)×C_3
    This is Proposition 2.1, imported from Tsang [7], specializing the Hopf-Galois construction to q=5; the paper's sketch argues the action α is trivial because X ≃ C_6 is abelian.
  • standard math PSU_5(64) is a finite non-abelian simple group
    Background fact used in Theorem 1.1 to conclude the socle is PSU_5(64), to use perfectness of the additive group, and to invoke the splitting criterion.
  • standard math Finite non-abelian simple groups are perfect
    Used in Section 4 to conclude the additive group PSU_5(64)×A_5 is perfect since both factors are non-abelian simple.

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Cite this review

Pith. "Pith review of A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group." pith.science (2026). https://pith.science/paper/PBQLIOYG

@misc{pith2026260714059,
  author       = {Pith},
  title        = {Pith review of: A Finite Skew Brace with Perfect Additive Group and Almost Simple Multiplicative Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PBQLIOYG}},
  note         = {Machine review of arXiv:2607.14059}
}
abstract

We construct a finite skew brace whose additive group is perfect and whose multiplicative group is non-perfect and almost simple. This gives an affirmative answer to Problem 20.109 in the twenty-first edition of the Kourovka Notebook. The additive and multiplicative groups of our example are \[ \PSU_5(64)\times A_5 \quad\text{and}\quad \Aut(\PSU_5(64)), \] respectively. The construction combines a skew brace with additive group \(A_5\) and multiplicative group \((C_5\rtimes C_4)\times C_3\), the splitting of the automorphism extension of \(\PSU_5(64)\), and a semidirect product of skew braces.

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Works this paper leans on

7 extracted references · 2 canonical work pages

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    Costantini, A

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    arXiv 2026

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    Lucchini, F

    A. Lucchini, F. Menegazzo and M. Morigi,On the existence of a complement for a finite simple group in its automorphism group, Illinois J. Math.47(2003), no.1–2,395–418. doi:10.1215/ijm/1258488162

    arXiv 2003

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    Smoktunowicz and L

    A. Smoktunowicz and L. Vendramin,On skew braces, with an appendix by N. Byott and L. Ven- dramin, J. Combin. Algebra2(2018), no.1,47–86. doi:10.4171/JCA/2-1-3

    doi:10.4171/jca/2-1-3 2018

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    Luigi Vanvitelli

    C. S. Y. Tsang,Non-abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension, Bull. Lond. Math. Soc.55(2023), no.5,2324–2340. doi:10.1112/blms.12860. 5 Massimiliano di Matteo Dipartimento di Matematica e Fisica Università degli Studi della Campania “Luigi Vanvitelli” viale Lincoln5, Caserta (Italy) e-mail: massimilia...

    work page doi:10.1112/blms.12860 2023

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