Skew braces of size $pq$

E. Acri; M. Bonatto

Skew braces of size $pq$

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For primes p>q, this paper classifies all skew braces whose additive group has order pq: only the trivial brace unless p≡1 mod q, in which case exactly 2q+2 braces exist.

desk verdict Clean, honest classification of skew braces of order pq; the new non-abelian case is done via explicit orbit computations, and the completeness rests transparently on Byott's published classification. read the letter →

arxiv 1908.03228 v3 pith:RSFPCKW4 submitted 2019-08-08 math.GR math.QAmath.RA

E. Acri , M. Bonatto This is my paper · ORCID

classification math.GRmath.QAmath.RA MSC 16T2520D4516T05

keywords skewbraceYang-BaxterequationHopf-Galoisextensionregularsubgroupholomorphsemidirectproductgroupoforderpqbi-skew

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

This paper completes the classification of all skew left braces whose additive group has order pq, where p>q are primes. It proves that when p is not congruent to 1 modulo q, the only skew brace of size pq is the trivial one. When p is congruent to 1 modulo q, it gives a complete list of 2q+2 skew braces up to isomorphism, with explicit operations for every brace in the list. The result matters because each skew brace yields set-theoretical solutions to the Yang–Baxter equation, and this settles one of the open classification problems in that area.

What carries the argument

The central object is the holomorph Hol(A)=A\rtimes\mathrm{Aut}(A) of a group A, along with the Guarnieri–Vendramin correspondence: skew braces with additive group A are in bijection with regular subgroups of Hol(A), and isomorphism classes correspond to orbits of these subgroups under conjugation by \mathrm{Aut}(A). The paper imports Byott's explicit classification of the regular subgroups of Hol(C) and Hol(M) for groups of order pq, then computes the Aut(A)-conjugacy orbit representatives. The operation formulas are obtained from the map \lambda_a(b)=-a+a\circ b, which records the image of the regular subgroup under the projection to \mathrm{Aut}(A). A separate construction shows that certain semidirect products with commuting automorphism images yield bi-skew braces.

What would settle it

Fix primes p>q with p\equiv 1\pmod q and directly enumerate all regular subgroups of Hol(Z_p\times Z_q) and Hol(Z_p\rtimes_g Z_q). The claim is false if any such subgroup is not conjugate to one of the families displayed in Lemmas 3.2, 3.5, 3.7, or 3.10, or if the number of Aut-conjugacy orbits differs from 2q+2.

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Extended reading notes

Core claim

The main theorem states that for primes p>q, if p\not\equiv 1\pmod q there is exactly one skew brace of size pq, the trivial one. If p\equiv 1\pmod q, there are exactly 2q+2 skew braces up to isomorphism, and the paper writes down their operations explicitly. The additive group is either the abelian group Z_p\times Z_q or the non-abelian semidirect product Z_p\rtimes_g Z_q, where g is a fixed element of order q modulo p. On the abelian side there are two braces (the trivial one and one non-trivial bi-skew brace); on the non-abelian side there are the trivial brace, one brace with |\ker(\$\lambda$)|=q, q-1 braces with |\ker(\$\lambda$)|=p, and q-1 braces with trivial kernel, each given by explicit formulas. The paper claims every skew brace of order pq appears exactly once in this list.

Load-bearing premise

The completeness of the final list rests entirely on Byott's classification of the regular subgroups of Hol(C) and Hol(M) for groups of order pq, quoted in Lemmas 3.2, 3.5, 3.7, and 3.10; if that classification missed any regular subgroup, the corresponding skew brace would be missing from the theorem.

Editorial extensions

If this is right

Every non-degenerate set-theoretic solution to the Yang–Baxter equation whose associated skew brace has size pq is realised by one of the listed braces, once the solution-to-skew-brace reduction is applied.
For p\not\equiv 1\pmod q, there are no nontrivial skew brace structures of order pq: any skew brace of that size is the trivial one.
For p\equiv 1\pmod q, the census size depends only on q, not on p: there are always 2q+2 braces, regardless of how large p is.
Among the 2q+2 braces, two are of cyclic additive type, and the remaining 2q are of non-abelian type, with the paper giving explicit bi-skew or non-bi-skew distinctions where relevant.
The explicit formulas allow any given skew brace of order pq to be checked against the list without searching through holomorphs.

Reading between the lines

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

The orbit computations are self-contained once Byott's regular-subgroup families are fixed, so the classification can be verified mechanically for any concrete pair p,q by enumerating conjugacy classes inside Hol(A).
Because the same regular subgroups classify Hopf–Galois structures, the orbit analysis here also yields a count of Hopf–Galois structures of degree pq, although the paper presents it only in the language of skew braces.
Specialising to q=2 gives a concrete prediction: for any odd prime p, there should be exactly 6 skew braces of order 2p; this is a quick finite check that a reader could carry out independently.
The paper leaves implicit which of the 2q+2 isomorphism classes admit a bi-skew or inverse brace structure; a natural follow-up is to label each class by these additional properties.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The short version: this paper answers Vendramin's problem for skew braces of size pq, and it does so in a workmanlike way. The genuinely new contribution is the classification of the non-abelian additive type braces: when p ≡ 1 mod q, there are 2q non-abelian braces, split into q-1 with kernel size p and q-1 with trivial kernel. Add the trivial braces and one cyclic-type brace, and you get the 2q+2 total. The explicit formulas in Theorems 3.9 and 3.12 are new, and the count matches the independent enumeration by Byott and Alabdali.

What the paper does well: it uses the holomorph correspondence seriously. Instead of guessing operations, the authors take Byott's regular subgroups of Hol(M) and compute orbits under conjugation by Aut(M). The orbit computations in Propositions 3.8 and 3.11 are explicit and easy to follow. The paper is clear about which parts come from Rump and which parts are new. The cyclic-type part is a reproof of known results, and the p not congruent to 1 case is a one-liner from Burnside's theorem, but both are needed for a complete statement.

Soft spots, in order of importance. First, the completeness of the classification is imported from Byott's 2004 paper on Hopf-Galois structures (Lemmas 3.2, 3.5, 3.7, 3.10). The paper does not reprove those classifications. That is not a defect if you trust Byott, and the authors are transparent, but anyone using the result should know that the entire list depends on that external source. Second, Theorems 3.6 and 3.12 contain 'straightforward to verify' checks of the brace axioms. In both cases the operation comes from a regular subgroup, so the axioms follow from the general correspondence, but the reader has to chase that through. I'd call this minor. I found no circularity, and the agreement with Byott-Alabdali is an independent cross-check.

Who is this for: people working on classification of skew braces and Hopf-Galois structures. It is a clean application of a known method to a small order family. It deserves a serious referee. No fatal problems that I can see. Have the referee focus on the completeness reduction and the orbit computations in the non-abelian case. I expect it to pass.

Recommendation: send to peer review.

Referee Report

0 major / 5 minor

Summary. The paper classifies all skew left braces of order pq, where p>q are primes, by combining Byott's classification of regular subgroups of holomorphs with the Guarnieri–Vendramin correspondence between skew braces and regular subgroups. The main theorem states that if p is not congruent to 1 modulo q, then the only skew brace of size pq is the trivial one; if p is congruent to 1 modulo q, then there are exactly 2q+2 skew braces up to isomorphism. These are described explicitly: two braces with abelian additive group Z_p × Z_q (one trivial, one bi-skew) and 2q braces with non-abelian additive group Z_p ⋊_g Z_q (one trivial, one with |ker λ|=q, q−1 bi-skew braces A_γ, and q−1 braces A_μ with ker λ=0). The proof proceeds by listing regular subgroups from Byott's work, computing their orbits under the action of the automorphism group, and then translating the orbit representatives into explicit brace operations using the holomorph correspondence.

Significance. If correct, this solves a natural and explicitly posed problem (Vendramin's Problem 2.15 in [24]). The result is clean and the explicit formulas for the operations are useful for further study of skew braces of small order. The paper also demonstrates a successful transfer of Hopf–Galois classification results into the skew-brace setting, and it includes a consistency check with the independent enumeration of Byott and Alabdali for squarefree order. The orbit computations are explicit and internally consistent, and the total orbit count matches the claimed 2q+2. The main external dependency is Byott's classification of regular subgroups, which is a published result; the paper's own contribution is the translation into brace language and the orbit analysis, which appears sound.

minor comments (5)

[Main Theorem; Theorems 3.9 and 3.12] The parametrization by γ and μ is ambiguous: the proofs define γ = (a+1)/a and μ = (d+1)/d as rational numbers, while the statements list '1<γ≤q' and '1<μ≤q'. Since g has multiplicative order q modulo p, the exponent should be understood modulo q, and γ and μ should be identified with the unique integer representatives in {2,...,q}, with q representing the residue 0. I recommend adding a sentence to clarify this point.
[Theorems 3.6 and 3.12] In the proofs of Theorems 3.6 and 3.12, the brace axioms are relegated to 'straightforward to verify' or 'it is easy to check'. Since the operations are derived from regular subgroups via Theorem 1.3, the brace property follows automatically from the regular subgroup construction; the authors could state this explicitly to avoid the impression of a gap.
[Proposition 3.11] The phrase 'as j g^{d+1}-1/(g-1) runs from 0 to p' is imprecise. The intended meaning is that as j runs through the residues modulo p, the coefficient runs through all p residues when d ≠ q−1, and is identically 0 when d = q−1; this should be phrased more carefully.
[Proposition 3.1] The term 'Burnside number' is used without definition. I suggest adding a parenthetical definition, e.g., gcd(n, φ(n)) = 1, to make the argument self-contained.
[References] Some cited works are preprints (e.g., [1] and [10]); the authors should update these references if they have appeared in final form.

Circularity Check

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No significant circularity found; the classification is derived from external, independent prior classifications.

full rationale

The paper's central theorem is a translation of Byott's 2004 classification of regular subgroups of Hol(C) and Hol(M) into skew-brace language via the Guarnieri–Vendramin holomorph correspondence (Theorem 1.3). The quoted lemmas (3.2, 3.5, 3.7, 3.10) are from Byott's independent work, not from the authors, and they do not already contain the paper's main result; the paper's own orbit computations (Propositions 3.3, 3.8, 3.11) then determine the isomorphism classes and produce explicit operations. There is no fitted parameter that is later called a prediction, no load-bearing self-citation, and no uniqueness theorem imported from the authors' own prior work. The agreement cited with Alabdali–Byott's enumeration is a consistency check, not an input. The completeness of the list does depend on Byott's external classification, but dependence on an independently published classification is not circularity. The claimed count 2q+2 follows from the orbit counts in the paper. Therefore no circular step can be exhibited from the text.

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

The central claim rests on external prior results: the holomorph correspondence (Guarnieri-Vendramin), Byott's regular subgroup classification for groups of order pq, Rump's cyclic brace classification, and the Burnside number result. No free parameters or invented entities are introduced; all constructed braces are explicit and checked against the regular subgroups.

assumptions (5)

standard math Isomorphism classes of skew braces over a fixed additive group A correspond bijectively to Aut(A)-conjugacy orbits of regular subgroups of Hol(A) (Theorem 1.3, citing [15]).
Used throughout Section 3 to convert regular subgroup orbits into skew brace isomorphism classes; accepted from the cited literature.
domain assumption Byott's classification of regular subgroups of Hol(C) and Hol(M) for groups of order pq is complete: families (11), (13), (15) and (18) exhaust all regular subgroups.
Quoted in Lemmas 3.2, 3.5, 3.7, 3.10 from [7]. Completeness of this external list is not reproved here and is load-bearing for the 'complete list' claim.
domain assumption For p not congruent to 1 mod q, pq is a Burnside number, so there is a unique skew brace of order pq ([22, Theorem A.8]).
Used in Proposition 3.1 to settle the p not congruent to 1 case without an explicit orbit enumeration.
domain assumption Aut(M) consists exactly of the maps phi_{i,j} of Lemma 2.1, from [14, Lemma 2.3].
Used to compute all conjugacy orbits in Sections 3.3; the description is taken from the cited paper.
standard math The automorphism group of the cyclic group of order pq has the presentation (7), with phi of order p-1 and psi of order q-1.
Elementary computation in Section 2; used to identify possible image sizes of lambda in the cyclic case.

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Pith. "Pith review of Skew braces of size $pq$." pith.science (2026). https://pith.science/paper/RSFPCKW4

@misc{pith2026190803228,
  author       = {Pith},
  title        = {Pith review of: Skew braces of size $pq$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSFPCKW4}},
  note         = {Machine review of arXiv:1908.03228}
}

abstract

We construct all skew braces of size $pq$ (where $p>q$ are primes) by using Byott's classification of Hopf--Galois extensions of the same degree. For $p\not\equiv 1 \pmod{q}$ there exists only one skew brace which is the trivial one. When $p\equiv 1 \pmod{q}$, we have $2q+2$ skew braces, two of which are of cyclic type (so, contained in Rump's classification) and $2q$ of non-abelian type.

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Works this paper leans on

24 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [24]

Vendramin

L. Vendramin. Problems on skew left braces. Advances in Group Theory and Applications , 7:15–37, 2019. (E. Acri, M. Bonatto) IMAS–CONICET and Universidad de Buenos Aires, P abellón 1, C iudad Uni- versitaria, 1428, Buenos Aires, Argentina E-mail address : eacri@dm.uba.ar E-mail address : marco.bonatto.87@gmail.com 9

work page 2019

[2] [1]

Skew Braces of Squarefree Order

A. Alabdali and N. P. Byott. Skew braces of squarefree ord er. Preprint. arXiv:1910.07814

work page Pith review arXiv 1910

[3] [2]

Bachiller

D. Bachiller. Classiﬁcation of braces of order p3. J. Pure Appl. Algebra , 219(8):3568–3603, 2015

work page 2015

[4] [3]

Bachiller

D. Bachiller. Counterexample to a conjecture about brac es. J. Algebra, 453:160–176, 2016

work page 2016

[5] [4]

Bachiller

D. Bachiller. Solutions of the Yang-Baxter equation ass ociated to skew left braces, with applications to racks. J. Knot Theory Ramiﬁcations , 27(8):1850055, 36, 2018

work page 2018

[6] [5]

Bachiller, F

D. Bachiller, F. Cedó, and E. Jespers. Solutions of the Ya ng-Baxter equation associated with a left brace. J. Algebra, 463:80–102, 2016

work page 2016

[7] [6]

N. P. Byott. Uniqueness of Hopf Galois structure for sepa rable ﬁeld extensions. Comm. Algebra , 24(10):3217– 3228, 1996

work page 1996

[8] [7]

N. P. Byott. Hopf-Galois structures on Galois ﬁeld exten sions of degree pq. J. Pure Appl. Algebra , 188(1-3):45–57, 2004

work page 2004