Simple Braces as Asymmetric Products

Definitions and mathematical statement

A left brace is a set $B$ with operations $+$ and $\circ$ such that $(B,+)$ is an abelian group, $(B,\circ)$ is a group, and \[ a\circ(b+c)=a\circ b-a+a\circ c\qquad(a,b,c\in B). \] Write $\lambda_a(b)=-a+a\circ b$. An ideal is an additive subgroup $I\leq (B,+)$ invariant under every $\lambda_a$ and normal in $(B,\circ)$. The brace is simple if its only ideals are $\{0\}$ and $B$. A brace is trivial if $a\circ b=a+b$ for all $a,b\in B$. A group $G$ is metabelian if its commutator subgroup $[G,G]$ is abelian. The condition on Sylow subgroups means that, for every prime $p\mid |G|$, every Sylow $p$-subgroup of $G$ is abelian.

For an explicit interpretation of the requested construction, the usual bilinear asymmetric product of two trivial braces can be described as follows. Let $T$ and $S$ be finite abelian groups, regarded as trivial braces. Let $b:T\times T\to S$ be symmetric and additive in each argument, and let $\alpha:(S,+)\to\operatorname{Aut}(T,+)$ be a group homomorphism such that \[ b(\alpha_s(t),\alpha_s(u))=b(t,u) \qquad(s\in S,\ t,u\in T). \] On $T\times S$, put \[ \begin{aligned} (t,s)+(u,v)&=(t+u,\ s+v+b(t,u)),\\ (t,s)\circ(u,v)&=(t+\alpha_s(u),\ s+v). \end{aligned} \] These operations describe the bilinear asymmetric-product convention used in the following formulation. The supplied record does not define asymmetric product, so agreement of this convention with the intended source construction requires review.

Question. Let $B$ be a nontrivial finite simple left brace whose multiplicative group $G=(B,\circ)$ is metabelian and has abelian Sylow subgroups. Must there exist $T,S,b,\alpha$ as above and a bijection $B\to T\times S$ preserving both brace operations?

The source qualification "nontrivial" is retained: the supplied record does not explicitly distinguish between excluding only the one-element brace and excluding every trivial brace. Accordingly, the precise range of this qualifier, as well as the asymmetric-product convention, remains subject to source review.

Short English statement

The question asks whether every nontrivial finite simple left brace with metabelian multiplicative group and abelian Sylow subgroups is isomorphic to an asymmetric product of two trivial braces. The draft makes the usual bilinear construction explicit, but its agreement with the intended source convention and the meaning of "nontrivial" require review. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004311$, title Simple Braces as Asymmetric Products, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. The record gives proposed year 2019 and names no proposer. Its supplied statement status is corrected_verified, with a recorded manual semantic, typographic, and LaTeX review; that label is not independent mathematical verification of this draft.
• Supplied report citation: Mini-Workshop: Algebraic Tools for Solving the Yang--Baxter Equation (2020), Oberwolfach Reports, pp. 3207--3242, DOI 10.4171/owr/2019/51. https://doi.org/10.4171/owr/2019/51. The supplied background describes the collection as drawn from workshop reports and their open-problem discussions.
• Supplied literature reference: New simple solutions of the Yang--Baxter equation. https://doi.org/10.1016/j.jpaa.2025.108025.
• Supplied literature reference: Constructing finite simple solutions. https://www.sciencedirect.com/science/article/abs/pii/S0001870821004072.
• Supplied literature reference: Oberwolfach Reports, skew braces problems. https://ems.press/content/serial-article-files/46831.
• Literature assessment dated exactly 2026-08-22: the supplied status is open. The supplied assessment says that asymmetric products construct important families of finite simple braces, but that no classification proves that every brace in the stated metabelian and Sylow setting has this form. This is supplied dated curation derived from the OWR record, not independently verified here; the record expressly requests independent verification of time-sensitive status and cited claims.