On skew braces and their ideals (with an Appendix by Agata Smoktunowicz)

TL;DR

This paper introduces combinatorial representations of finite skew braces, creates a database of small skew braces, and explores their ideal-related properties, providing new insights and raising questions in the theory.

Contribution

It develops a novel combinatorial approach to finite skew braces and constructs a comprehensive database to study their ideal structures and properties.

Findings

Database of small skew braces created

Analysis of ideals, prime and semiprime ideals conducted

Insights into radicals and solvability of skew braces obtained

Abstract

We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions.