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

“…A statement similar to that of Theorem 4.21 also holds if the concept of subbrace is replaced by that of (strong) left-ideal; we leave the details to the reader but we just note that, for example, we need to replace the fact that the brace B locally satisfies the maximal condition on subbraces by the request that every finite subset of B is contained in a (strong) left-ideal that satisfies the maximal condition on (strong) left-ideals. All these statements generalize Lemma 5.10 of [4].…”

“…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.…”

“…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.…”

Finite skew braces of square-free order and supersolubility

“…A statement similar to that of Theorem 4.21 also holds if the concept of subbrace is replaced by that of (strong) left-ideal; we leave the details to the reader but we just note that, for example, we need to replace the fact that the brace B locally satisfies the maximal condition on subbraces by the request that every finite subset of B is contained in a (strong) left-ideal that satisfies the maximal condition on (strong) left-ideals. All these statements generalize Lemma 5.10 of [4].…”

“…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.…”

“…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.…”

Commuting probability of skew left braces

“…Now using the fact that θ is an isomorphism and the diagram (4.6) commutes, we get Note that Γ 2 (A/ Ann(A)) = Γ 2 (A) Ann(A)/ Ann(A). Now by the third isomorphism theorem of skew braces [2], we get…”

Skew brace extensions, second cohomology and complements

“…We write B ≤ A to denote that B is a sub-skew brace of the skew brace A, and B ⊴ A to denote that B is an ideal of A. The quotient skew brace A/I of A by A will be denoted by Ā. Isomorphism theorems for skew braces were established in [5], which we'll use without further reference. The category of all skew left braces will be denoted by SB.…”

Finite skew braces of square-free order and supersolubility

“…A statement similar to that of Theorem 4.21 also holds if the concept of subbrace is replaced by that of (strong) left-ideal; we leave the details to the reader but we just note that, for example, we need to replace the fact that the brace B locally satisfies the maximal condition on subbraces by the request that every finite subset of B is contained in a (strong) left-ideal that satisfies the maximal condition on (strong) left-ideals. All these statements generalize Lemma 5.10 of [4].…”

“…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.…”

“…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.…”

Commuting probability of skew left braces

“…Now using the fact that θ is an isomorphism and the diagram (4.6) commutes, we get Note that Γ 2 (A/ Ann(A)) = Γ 2 (A) Ann(A)/ Ann(A). Now by the third isomorphism theorem of skew braces [2], we get…”

Skew brace extensions, second cohomology and complements

“…We write B ≤ A to denote that B is a sub-skew brace of the skew brace A, and B ⊴ A to denote that B is an ideal of A. The quotient skew brace A/I of A by A will be denoted by Ā. Isomorphism theorems for skew braces were established in [5], which we'll use without further reference. The category of all skew left braces will be denoted by SB.…”

Finite skew braces of square-free order and supersolubility

“…A statement similar to that of Theorem 4.21 also holds if the concept of subbrace is replaced by that of (strong) left-ideal; we leave the details to the reader but we just note that, for example, we need to replace the fact that the brace B locally satisfies the maximal condition on subbraces by the request that every finite subset of B is contained in a (strong) left-ideal that satisfies the maximal condition on (strong) left-ideals. All these statements generalize Lemma 5.10 of [4].…”

“…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.…”

“…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.…”

Commuting probability of skew left braces

“…Now using the fact that θ is an isomorphism and the diagram (4.6) commutes, we get Note that Γ 2 (A/ Ann(A)) = Γ 2 (A) Ann(A)/ Ann(A). Now by the third isomorphism theorem of skew braces [2], we get…”

Skew brace extensions, second cohomology and complements

“…We write B ≤ A to denote that B is a sub-skew brace of the skew brace A, and B ⊴ A to denote that B is an ideal of A. The quotient skew brace A/I of A by A will be denoted by Ā. Isomorphism theorems for skew braces were established in [5], which we'll use without further reference. The category of all skew left braces will be denoted by SB.…”