A Jordan–Hölder theorem for skew left braces and their applications to multipermutation solutions of the Yang–Baxter equation

Abstract

Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.

Finite skew braces of square-free order and supersolubility

“…Consider the semidirect product G of B and C with respect to this action (we use multiplicative notation in G). Then G turns out to be a trifactorised group as it possesses a subgroup 𝐷 = 𝑎 8 𝑥, 𝑎 6 𝑦, 𝑏𝑧, 𝑎 3 𝑡 such that 𝐷 ∩ 𝐶 = 𝐷 ∩ 𝐵 = {1}, 𝐷𝐶 = 𝐵𝐷 = 𝐺. Thus, there is a bijective 1-cocycle 𝛿 : 𝐶 −→ 𝐵 with respect to 𝜆 given by Table 3.…”

“…As shown in [6], a great deal of structural information is provided by the chief factors of a brace. Recall that if B is brace and 𝐽 < 𝐼 are ideals of B such that 𝐼/𝐽 is a minimal ideal of 𝐵/𝐽, then 𝐼/𝐽 is said to be a chief factor of B.…”

“…Note also that if L is a left-ideal of B and , then is a left-ideal of B , and is an ideal of B ; in particular, . The following (here, and denote the centres of and , respectively) are further relevant examples of ideals of B playing key roles in the study of nilpotency of braces (see [6] or [17]).…”

“…A further nice feature of supersoluble skew left braces is that they have a largest centrally nilpotent ideal (the Fitting ideal ), an ideal that does not exist even in arbitrary finite skew left braces (see [4]) – in fact, a centrally nilpotent ideal I of a supersoluble skew left brace B has a finite chain of ideals whose factors are central in I (see Theorem 3.36). In the supersoluble environment, the Fitting ideal has finite index (see Theorem 3.42), and on some occasions its index is even a power of (see Theorem 3.45).…”

“…Significant examples of such a fruitful approach can be found in the analysis of two of the most studied properties of solutions since their introduction in [20]: multipermutation solutions, that is, those solutions that can be retracted into the trivial solution over a singleton after finitely many identification steps, and indecomposable solutions, or those solutions that can not be decomposed in a disjoint union of two proper solutions. Concretely, nilpotency of skew left braces has been introduced to deal with the former ones (see [6,17,26,30], among others), and simplicity of skew left braces play a key role in the study of indecomposable solutions (see [13,14,15,39]). Moreover, in [7] solubility of skew left braces has been studied as an opposite class of simplicity that allows multidecomposability of solutions.…”