On a Cauchy theorem for finite skew braces

TL;DR

This paper proves a Cauchy theorem for certain classes of finite skew braces, advancing the classification of these algebraic structures and providing new structural insights.

Contribution

It establishes a positive Cauchy theorem for finite two-sided and bi-skew braces, a significant step in their structural classification.

Findings

Cauchy theorem holds for finite two-sided skew braces

Cauchy theorem holds for finite bi-skew braces

Structural consequences for skew brace classification

Abstract

One of the major problems in the structural theory of skew braces consists in the classification of skew braces of finite order up to isomorphism. In this light, the open question of the existence of a Cauchy theorem for finite skew braces is of great interest. We prove a positive answer for the classes of finite two-sided skew braces and bi-skew braces. Consequences and related structural results are also outlined.