On skew braces and their ideals

Konovalov, Alexander; Smoktunowicz, Agata; Vendramin, L.

doi:10.1080/10586458.2018.1492476

Cited by 41 publications

(41 citation statements)

References 41 publications

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“…Konovalov, Smoktunowicz and Vendramin introduced prime left braces (see [20]). Recall that in a left brace B, the operation * is defined by a * b = ab−a−b, and for ideals I and J, I * J denotes the additive subgroup of B generated by {a * b | a ∈ I, b ∈ J}.…”

Section: Finite Prime Left Braces With Multiplicative Agroupmentioning

confidence: 99%

“…It is clear that every non-trivial simple left brace is prime. In [20,Question 4.4] the following question is posed: do there exist finite prime non-simple left braces? Using the construction of left braces as introduced in Section 3 we will show that the answer to this question is affirmative.…”

Section: Finite Prime Left Braces With Multiplicative Agroupmentioning

confidence: 99%

“…Our construction requires special elements of the Sylow p-subgroups of the orthogonal groups over finite fields of characteristic relatively prime to p. The existence and limitations concerning such elements are discussed in Section 4. A problem on prime braces, recently proposed in [20], Question 4.3, is solved in Section 5. Finally, in Section 6, we construct another new family of finite simple left braces using the technique of Section 3.…”

Section: Introductionmentioning

confidence: 99%

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An abundance of simple left braces with abelian multiplicative Sylow subgroups

2020

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Braces were introduced by Rump to study involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation. A constructive method for producing all such finite solutions from a description of all finite left braces has been recently discovered. It is thus a fundamental problem to construct and classify all simple left braces, as they can be considered as building blocks for the general theory. This program recently has been initiated by Bachiller and the authors. In this paper we study the simple finite left braces such that the Sylow subgroups of their multiplicative groups are abelian. We provide several new families of such simple left braces. In particular, they lead to the main, surprising result, that shows that there is an abundance of such simple left braces.

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Section: Finite Prime Left Braces With Multiplicative Agroupmentioning

confidence: 99%

Section: Finite Prime Left Braces With Multiplicative Agroupmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An abundance of simple left braces with abelian multiplicative Sylow subgroups

2020

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“…Example 2.10. There exists a unique simple left brace of size 72, see [12,Remark 4.5] and [38,Proposition 4.3]. The multiplicative group of this left brace is isomorphic to A 4 × S 3 and therefore all of its Sylow subgroups are abelian.…”

Section: 1mentioning

confidence: 99%

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

Acri

Lutowski

Vendramin

2019

Int. J. Algebra Comput.

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Using Bieberbach groups we study multipermutation involutive solutions to the Yang-Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right p-nilpotent skew braces. The theory of left p-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.

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“…They also have applications in knot theory due to connections with biquandles and racks [1,12]. Some algebraic aspects of skew braces are studied in [5,8,12]. A big list of problems concerning skew braces is collected in [13].…”

Section: Introductionmentioning

confidence: 99%

Connections between properties of the additive and the multiplicative groups of a two-sided skew brace

Nasybullov

2019

Journal of Algebra

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We study relations between the additive and the multiplicative groups of a two-sided skew brace. In particular, we prove that if the additive group of a two-sided skew brace is finite solvable (respectively, finitely generated nilpotent, finitely generated residually nilpotent, finitely generated residually finite), then the multiplicative group of this skew brace is solvable (respectively, solvable, residually solvable, residually finite). Also we prove that if the multiplicative group of a two-sided skew brace is nilpotent of nilpotency class k, then the additive group of this skew brace is solvable of class at most 2k. The letter result generalizes the result from [12, Theorem A.9] which says that if the multiplicative group of a finite skew brace is abelian, then the additive group of this skew brace is solvable.In addition we solve two problems (Problem 19.49 and Problem 19.90(a)) concerning skew braces which are formulated in the Kourovka notebook.

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On skew braces and their ideals

Cited by 41 publications

References 41 publications

An abundance of simple left braces with abelian multiplicative Sylow subgroups

An abundance of simple left braces with abelian multiplicative Sylow subgroups

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

Connections between properties of the additive and the multiplicative groups of a two-sided skew brace

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