A simple brace outside asymmetric products of two trivial braces

Abstract

We give a negative answer to Cedó's Problem 9 in Oberwolfach Report 51/2019. An explicit left brace of order 20 000 is nontrivial and simple, has metabelian multiplicative group with abelian Sylow subgroups, and is not an asymmetric product of two trivial braces. The obstruction is intrinsic: every such two-factor product has zero triple star products within each additive Sylow subbrace, whereas the constructed brace has (x*x)*x=x³≠0 in its 5-primary subbrace. The obstruction covers the full admissible symmetric-cocycle definition, not only a chosen bilinear construction family. All operations and the simplicity argument are given directly; exact verification programs and independent adversarial audits accompany the note.

The idea in plain English

A brace is a set with two compatible group operations. Such structures connect algebra with the Yang–Baxter equation. Here “simple” means that there is no nonzero proper ideal compatible with both operations; it does not mean that the multiplicative group is simple.

Cedó’s 2019 question asks whether all finite nontrivial simple left braces with two specified restrictions on their multiplicative group come from one construction: an asymmetric product of two trivial braces. The restrictions are that the group is metabelian and that every one of its Sylow subgroups is abelian.

The paper presents a 20,000-element counterexample. Its distinguishing feature is an internal calculation that survives every change of coordinates: a triple star product is nonzero in the 5-primary part, whereas the proposed two-factor construction forces all such products to vanish there.

The precise result

Theorem 1 specifies a left brace B with additive group C₅⁴ × C₂⁵ and multiplicative group (C₅³ ⋊ C₂) × (C₂⁴ ⋊ C₅), with the two actions explicitly given. It is nontrivial and simple, and its multiplicative group is nonabelian and metabelian with all Sylow subgroups abelian.

Lemma 2 is a presentation-independent obstruction. In any finite asymmetric product of two trivial braces with abelian multiplicative Sylow subgroups, (a∗b)∗c = 0 whenever a, b and c lie in one additive Sylow subbrace. The constructed brace instead has (x∗x)∗x = x³ ≠ 0 in its 5-primary subbrace, ruling out every such presentation.

Why the result holds

The construction combines the truncated ring xF₅[x]/(x⁵) with a four-dimensional binary space and one extra binary coordinate. A finite logarithm change of coordinates identifies the multiplicative group explicitly; no infinite series or division by five is used.

The simplicity argument starts with an arbitrary nonzero ideal and forces it to contain generators of both primary parts. It does not assume that ideals are coordinate subspaces.

The obstruction begins with a normalized symmetric cocycle in the general asymmetric-product definition. Brace compatibility forces biadditivity in the two-trivial-factor setting. The abelian multiplicative Sylow condition then forces the indicated primary triple products to vanish.

Scope and limitations

The result is one explicit counterexample of order 20,000. It makes no smallest-order claim and does not classify all finite simple braces.

The excluded presentations have exactly two trivial factors. The paper does not exclude asymmetric products with nontrivial factors, more elaborate products, or general matched-product descriptions.

The universal obstruction assumes a finite left brace and abelian multiplicative Sylow subgroups. It concerns the left-associated product (a∗b)∗c with all three elements in the same additive primary part; it does not assert that every mixed-primary triple product vanishes.

The paper is an unrefereed, extensively AI-assisted preprint. Its historical-priority discussion is explicitly bounded, and it does not claim a new general construction formalism.

Status and AI use

AI tools were used extensively in construction, proof development, computation, verification, literature research and writing. AI review does not constitute external human peer review or proof-assistant certification.

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Collaborate on this result

I welcome substantive collaboration from researchers in brace theory, finite groups and the Yang–Baxter equation on independent verification, comparison with earlier constructions and preparation for journal review.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). A simple brace outside asymmetric products of two trivial braces (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23441906

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Background and prior work

• Cedó — Finite simple left braces: constructions and problems: Problem 9, printed page 3228, gives the exact finite, nontrivial, simple, metabelian and abelian-Sylow question addressed here.
• Catino, Colazzo and Stefanelli — Regular subgroups of the affine group and asymmetric product of radical braces: Introduces the asymmetric-product framework; the present note does not claim that construction formalism as new.
• Bachiller, Cedó, Jespers and Okniński — Asymmetric product of left braces and simplicity: Theorem 3.1 states the general normalized symmetric-cocycle construction in left-brace notation.
• Cedó, Jespers and Okniński — Every finite abelian group is a subgroup of the additive group of a finite simple left brace: Theorem 2.2 gives the asymmetric-product and lambda formulas; the discussion distinguishes a twisted additive group from an ordinary direct product, especially at the prime two.
• Bachiller — Extensions, matched products, and simple braces: Matched-product methods and additive-primary decompositions are established prior tools; the note supplies its own explicit operations and all-ideal argument.