On Yang-Baxter groups

Finite skew braces of square-free order and supersolubility

“…We now briefly recall the nilpotency concepts we need, and some of their properties; we refer the interested reader to [4], [8], [12], [16], [17], [30], [37] for more information on the subject. Let B be a brace, and let 𝐶, 𝐷 be subsets of B.…”

“…for every 𝑛 ∈ N. Then 𝐵 = 𝜁 𝑚 (𝐵) if and only if Γ 𝑚+1 (𝐵) = {0}. Although central nilpotency is the strongest nilpotency concept for braces, it has been shown in [4] that the sum of two centrally nilpotent ideals need not be centrally nilpotent, even in finite braces: The problem is that a centrally nilpotent ideal I of a brace B need not have a finite chain of ideals of B that are central in I. To avoid this problem, we gave the following definition in [4]: An ideal I of a brace B is B-centrally nilpotent if there exists a finite chain of ideals of B…”

“…This is called the upper B-central series of I and is defined as follows: Put 𝜁 0 (𝐼) 𝐵 = {0}, 𝜁 𝜆 (𝐼) 𝐵 = 𝛼<𝜆 𝜁 𝛼 (𝐼) 𝐵 for every limit ordinal 𝜆, and 𝜁 𝛼+1 (𝐼) 𝐵 /𝜁 𝛼 (𝐼) 𝐵 to be the largest ideal of 𝐵/𝜁 𝛼 (𝐼) 𝐵 contained in 𝜁 𝐼/𝜁 𝛼 (𝐼) 𝐵 for every ordinal number 𝛼; in particular, 𝜁 (𝐼) 𝐵 = 𝜁 1 (𝐼) 𝐵 is the largest ideal of B contained in 𝜁 (𝐼). Of course, I is B-centrally nilpotent if and only if 𝜁 𝑛 (𝐼) 𝐵 = 𝐼 for some 𝑛 ∈ N. Now, if I and J are B-centrally nilpotent ideals of B, then 𝐼 + 𝐽 is B-centrally nilpotent (see [4], Theorem 5.3); the sum of all B-centrally nilpotent ideals of B is the Fitting ideal Fit(𝐵) of B. We recall also that if 𝐼 = 𝜁 𝜇 (𝐼) 𝐵 for some arbitrary ordinal number 𝜇, then I is said to be B-hypercentral; thus, I is B-hypercentral if and only if it has an ascending chain of ideals of B with I-central factors.…”

“…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. These significant examples show that the classification of (finite) skew left braces undoubtedly provides a powerful tool to classify the solutions of the YBE (to some extent, at least).…”

“…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 2 (see Theorem 3.45).…”