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A Counterexample to Byott's Conjecture for Finite Skew Braces

T0 review · 1 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read There exists a finite skew brace whose additive group is solvable but whose multiplicative group is not; the multiplicative group has a quotient isomorphic to PSL2(7), so Byott's conjecture—that regular subgroups of Hol(N) for solvable N ar

desk verdict First counterexample to Byott's conjecture, with an explicit finite skew brace; the construction is credible and the residual risk is a handful of unprinted GAP-verified tables. read the letter →

arxiv 2607.22795 v1 pith:W5PJFYPU submitted 2026-07-24 math.GR math.RA

Massimiliano Di Matteo , Maria Ferrara , Marco Trombetti This is my paper · ORCID
classification math.GRmath.RA MSC 16T2520D1012F10
keywords skewbraceByott'sconjectureHopf–GaloisstructureregularsubgroupholomorphbijectivecocyclesolvablegroupPSL2(7)
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 proves Byott's conjecture false by constructing an explicit finite skew brace (B,+,∘) with (B,+) solvable and (B,∘) insoluble. Specifically, (B,+) is the opposite of an iterated semidirect product F_7^8 ⋊ (F_3^7 ⋊ F_2^3), while (B,∘) contains a normal solvable subgroup H whose quotient is GL3(2) ≃ PSL2(7), a simple group of order 168. Because a skew brace encodes the same data as a regular subgroup of the holomorph and as a Hopf–Galois structure, the example also disproves the conjecture in those formulations. The construction is fully explicit: every map is given by matrices and tables, and the finite verifications were done computationally.

What carries the argument

The construction rests on the cocycle c_g = (β_P(g), β_W(g), π(g)) in the opposite group N = N0^op, together with λ_g = Inn_{−c_g} ϑ_g. Here π(g) is the translation part of the unique affine lift bg = (g, π(g)) of g in an affine copy bL ≃ PSL2(7) of GL3(2); the functions β_P and β_W are defined from the stabilizer decomposition σ(g,x) = t_{g⋆x}^{−1} g t_x = U^{m(g,x)} Y^{k(g,x)} of the affine action g⋆x = gx + π(g), where F = ⟨U,Y⟩ ≃ C7⋊C3 is the stabilizer of e0. The spaces P = F7^E and W = {zero-sum functions E→F3} carry twisted actions with multiplier 2^{k(g,x)}, chosen so that β_P and β_W are 1-cocycles; the opposite-group law reverses the semidirect order so that c_gh = c_g + λ_g(c_h) h

What would settle it

Recompute the transversal in Table 1 and the exponents m and k in Table 2 from the displayed matrices A and B, then verify for the generators A and B that β_P(AB) = β_P(A) + ρ_P(A)β_P(B) and similarly for β_W; any mismatch would make the cocycle identities (15) or (16) fail and destroy the brace structure.

Watch

Extended reading notes

Core claim

The central discovery is one finite skew brace with solvable additive group and non-solvable multiplicative group. The additive group N is the opposite of N0 = P ⋊ (W ⋊ E), where P = F7^{E} is the vector space of all functions on E = F2^3, W is the zero-sum hyperplane of F3^{E}, and E is the underlying vector space. The multiplicative group is a semidirect product G = H ⋊ L with L = GL3(2) ≃ PSL2(7) and H a subgroup of N of index 168. The cocycle c_g = (β_P(g), β_W(g), π(g)) is built from two 1-cocycles induced by an affine copy of PSL2(7) in AGL(E), and the opposite-group twist is what orients the cocycle identity correctly.

Load-bearing premise

The load-bearing premise is that the finite computations listed in Tables 1 and 2 and the matrix identities in Lemma 2.1 are error-free: the paper says they were verified computationally but does not display all the arithmetic.

Editorial extensions

If this is right

  • The regular-subgroup version of Byott's conjecture fails: there is a regular embedding of the insoluble group G into Hol(N) for a finite solvable group N.
  • In Hopf–Galois theory, an insoluble finite Galois group can admit a Hopf–Galois structure of solvable type.
  • The known affirmative cases — nilpotent additive group, orders not divisible by 3, two-sided braces, orders up to 2000 — remain true, but none of them extends to all solvable additive groups.
  • The same skew brace gives a finite set-theoretic solution of the Yang–Baxter equation whose 'base' group is solvable and whose 'derived' group is not.

Reading between the lines

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

  • The opposite-group trick may generalize: any affine action of a non-solvable group on a finite vector space with a suitable flag of stabilizers could yield similar cocycles, and the minimal counterexample might be far smaller than the ~10^11-order example presented here.
  • Since the additive group is a semidirect product of vector spaces, this construction suggests that restricting the additive group to be a p-group or a direct product of elementary abelian groups will not recover Byott's conjecture.
  • One could test whether the quotient PSL2(7) can be replaced by other small non-abelian simple groups, e.g., by finding analogous stabilizer subgroups C_q ⋊ C_p inside an affine general linear group with the right exponent relations.
  • A useful public artifact would be a complete verification script (or expanded tables) for all 168 cocycle identities; the paper's computation is convincing but not independently checkable from the printed text alone.
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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

1 major / 5 minor

Summary. The paper constructs an explicit finite skew brace (B,+,∘) with soluble additive group and insoluble multiplicative group. The additive group is the opposite of a soluble group N0 = P ⋊ (W ⋊ E), with P ≅ F_7^8, W ≅ F_3^7, E ≅ F_2^3, while the multiplicative group is a semidirect product H ⋊ L with L = GL_3(2) ≅ PSL_2(7), giving an insoluble quotient. The construction proceeds by embedding an affine copy of GL_3(2) into AGL(E), building a transversal and exponent tables, defining cocycles β_W and β_P, forming the group N0, passing to the opposite group, and transporting a semidirect-product law to N. The main theorem is stated as a counterexample to Byott's conjecture, and the paper also derives a regular embedding into Hol(N).

Significance. If correct, this is a significant result: it disproves Byott's conjecture that every regular subgroup of the holomorph of a finite soluble group is soluble. The construction is fully explicit and the main structural steps — the cocycle identities (10)–(11) and (15)–(16), the automorphism property of the ϑ_g, and the coset partition underlying the bijection b — are proved from the definitions. The paper also connects the counterexample to Hopf–Galois structures. A notable strength is that the proof is coordinate-based and largely checkable by direct multiplication. The main weakness is that several load-bearing finite computations are reported as checked in GAP but not fully displayed; I spot-checked some of them and found no error.

major comments (1)
  1. [§2, Tables 1–2 and the GAP-checked assertions (pp. 4–8)] The central claim depends on the finite data in Table 1 (the transversal t_x) and Table 2 (the exponent functions m and k for A and B), as well as on the matrix identities in Lemma 2.1 and the stabilizer factorization F=⟨U,Y⟩. These data feed directly into Lemma 2.2, Proposition 2.3, and Lemma 2.6; for example, a single incorrect entry in Table 2 would break the cocycle identities (10)–(11), the conclusion β_W(g)∈W in Proposition 2.3, and the partition (33). The text verifies only two sample entries of Table 2 and states that 'every other entry is obtained by the same matrix multiplication'; the GAP code is not supplied. I independently recomputed several entries (t_e3, t_e5, t_e7, bY, bB^2) and found them internally consistent, so I do not suspect an error; nevertheless, the audit trail should be completed by including the code or a full computation appendix.
minor comments (5)
  1. [Lemma 2.1, Eq. (3)] The presentation PSL2(7)=⟨x,y | x²=y³=(xy)⁷=[x,y]⁴=1⟩ is called standard but no reference or justification is given. Since the proof of Lemma 2.1 depends on it, please add a reference or a one-line justification.
  2. [§2, after Table 2] The sentence 'Every other entry is obtained by the same matrix multiplication and unique reduction to U^mY^k' would be more useful with at least one additional fully worked example, especially for an entry with k≠0.
  3. [§2, after Eq. (14)] The coordinate-vector identification for functions E→F_q is introduced only after β_W and β_P are defined. It would be clearer to introduce this identification before displaying the vectors (0,0,0,1,2,2,0,1) and (1,2,0,1,0,1,0,1).
  4. [Lemma 2.6, p. 11] The proof that the 168 triples q(c_g) are pairwise distinct is compressed: it relies on the uniqueness of g=t_xU^mY^k, but that uniqueness is justified only by a counting argument. This is correct, but because the partition (33) is foundational for the bijection b, the counting argument should be spelled out explicitly.
  5. [Throughout] The symbol F is used both for the stabilizer subgroup of L and for the fields F_2,F_3,F_7. This is conventional but potentially confusing; consider using blackboard-bold or a different letter for the stabilizer.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is explicit and the target properties are proved from the displayed matrices, tables, and group laws, with no fitted parameter or self-citation used as the load-bearing premise.

full rationale

The paper's derivation chain is fully explicit: a finite affine group bL is defined by concrete matrices A and B; Lemma 2.1 verifies bL ≅ PSL2(7) by direct multiplication and the standard presentation. The transversal Table 1, the exponent Table 2, and the functions beta_W, beta_P are all defined from explicit matrix computations and are used to verify the cocycle identities (15) and (16). The semidirect products N0, the automorphisms theta_g, the elements c_g, the subgroup H, and the bijection b are all constructed from these data. The skew-brace identity (1) is then proved directly from the cocycle identity (39) and the fact that each lambda_a is an automorphism of (N,+). The insolubility of (N,∘) is established by exhibiting a normal subgroup H with (N,∘)/H ≅ GL3(2) ≅ PSL2(7), computed from the explicit construction, not assumed. There is no step where an input is defined in terms of the conclusion, no parameter fitted to the target data and then renamed a prediction, and no load-bearing reliance on the authors' prior results; references to Byott's conjecture and to [3] only frame the problem and do not enter the proof. The only unshown parts are the finite arithmetic checks said to have been performed in GAP; even if those checks were incomplete, that is a verification or correctness gap, not circular reasoning. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No fitted parameters or new postulated entities appear. The construction is explicit finite group theory; the only non-proven-in-text items are finite matrix computations, which the authors state were verified in GAP.

assumptions (3)
  • standard math The presentation PSL2(7)=⟨x,y | x^2=y^3=(xy)^7=[x,y]^4=1⟩ is valid and PSL2(7) is simple.
    Used in Lemma 2.1 to identify the lifted affine group with L=GL3(2)≅PSL2(7). This is a standard fact in finite group theory.
  • ad hoc to paper The finite computations in Table 1 (transversal t_x), Table 2 (exponents m and k), and the matrix identities for b_Y, U, and Y are correct.
    The proof describes them as 'direct multiplication' and says they were checked in GAP, but it does not display full computations. If an entry is incorrect, the cocycle identities and the final brace construction fail.
  • standard math The dictionary among regular subgroups of Hol(N), bijective 1-cocycles, and skew braces is correct as used in Section 1.
    The paper relies on this equivalence for motivation and for the final holomorph formulation; the direct construction of b and λ also proves the needed direction in this specific case.

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Pith. "Pith review of A Counterexample to Byott's Conjecture for Finite Skew Braces." pith.science (2026). https://pith.science/paper/W5PJFYPU

@misc{pith2026260722795,
  author       = {Pith},
  title        = {Pith review of: A Counterexample to Byott's Conjecture for Finite Skew Braces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5PJFYPU}},
  note         = {Machine review of arXiv:2607.22795}
}
abstract

We construct a finite skew brace with soluble additive group and insoluble multiplicative group. The multiplicative group has a quotient isomorphic to $\PSL_2(7)$, and hence the example disproves Byott's conjecture.

Discussion (0). Continue with ORCID to comment.

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