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Finite skew braces of square-free order and supersolubility

Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Maria Ferrara, Vicent Pérez-Calabuig, Marco Trombetti

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 non-degenerate 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.

Finite skew braces of square-free order and supersolubility

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 many relevant skew brace-theoretical properties are easier to identify: for example, a centrally nilpotent ideal of is -centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, has finite multipermutational level if and only if is nilpotent. Given a finite presentation of the structure skew brace associated with a finite non-degenerate solution of the Yang--Baxter Equation (YBE), there is an algorithm that decides if 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.
Paper Structure (5 sections, 82 theorems, 67 equations, 4 tables)

This paper contains 5 sections, 82 theorems, 67 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Supersoluble braces
  4. Hypercyclic and locally supersoluble braces
  5. Acknowledgements

Key Result

Lemma 3.5

Let $B$ be a brace, and let $G=(B,+)\rtimes_\lambda(B,\boldsymbol{\cdot})$. If $B$ is supersoluble, then $G$ is supersoluble. In particular, $(B,+)$ and $(B,\boldsymbol{\cdot})$ are supersoluble groups.

Theorems & Definitions (101)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Corollary 3.9
  • Theorem 3.10: tutti23-2, The-o-rem 3.4
  • ...and 91 more