On the Sylow Theorem for Skew Braces

TL;DR

This paper extends Sylow theorem concepts to various classes of finite skew braces, establishing conditions under which the theorem holds and proving related Hall-type theorems.

Contribution

It generalizes Sylow theorem applicability to multiple classes of skew braces, including two-sided, bi-skew, and nilpotent types, with new Hall-type results.

Findings

Sylow theorem holds for two-sided skew braces

Sylow theorem applies to bi-skew and right nilpotent skew braces

Hall-type theorems are established for specialized skew brace classes

Abstract

We discuss the (first) Sylow theorem for certain classes of finite skew braces, proving it to hold true when the skew brace is two-sided, bi-skew, right nilpotent, $\lambda$-homomorphic or supersoluble. We also show it to hold true for soluble skew braces that are left-nilpotent, and address a number of more specialized settings, proving general Hall-type theorems.