Byott's conjecture on solubility of finite skew braces

Byott's conjecture asks whether a finite skew brace with soluble additive group must have soluble multiplicative group. It arose from Byott's work on Hopf–Galois structures and is listed as Problem 19.91 in the Kourovka Notebook.

Known results

• The conjecture holds for bi-skew braces (Bachiller, Smoktunowicz, and Vendramin, 2023).
• It holds when the brace order is not divisible by $3$ (Smoktunowicz and Vendramin, 2020).
• Known affirmative cases include two-sided braces, nilpotent additive groups, cube-free orders, and all orders at most $2000$.
• A minimal counterexample, if one exists, cannot have simple multiplicative group (Smoktunowicz and Vendramin, 2020).

2026 counterexample

The preprint A Counterexample to Byott's Conjecture for Finite Skew Braces claims that Di Matteo, Ferrara, and Trombetti construct a finite skew brace with soluble additive group but insoluble multiplicative group; the latter has a quotient isomorphic to $\operatorname{PSL}_2(7)$. The paper gives a coordinate proof and reports a GAP check, but this claimed refutation is not independently verified in the retrieved sources.

Current status (as of September 2026): The general conjecture is claimed false by a 2026 counterexample, while the bi-skew and other restricted cases are settled; independent verification of the counterexample is not recorded.