Finite skew braces of square-free order and supersolubility

Author:

Ballester-Bolinches A.ORCID,Esteban-Romero R.ORCID,Ferrara M.ORCID,Pérez-Calabuig V.ORCID,Trombetti M.ORCID

Abstract

Abstract The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers and that in an arbitrary supersoluble skew brace B many relevant skew brace-theoretical properties are easier to identify: For example, a centrally nilpotent ideal of B is B-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, B has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite nondegenerate solution of the Yang–Baxter equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on.

Publisher

Cambridge University Press (CUP)

Reference42 articles.

1. [4] Ballester-Bolinches, A. , Esteban-Romero, R. , Ferrara, M. , Pérez-Calabuig, V. and Trombetti, M. , ‘Central nilpotency of left skew braces and solutions of the Yang–Baxter equation’, Preprint, 2023, arxiv:2310.07474v2.

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3. Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks

4. Classes of finite groups and their properties

5. Triply Factorised Groups and the Structure of Skew Left Braces

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