Skew Braces of Squarefree Order

TL;DR

This paper classifies and counts skew braces of squarefree order, linking their structure to Hopf-Galois theory, and provides explicit enumeration for cases involving products of three distinct primes.

Contribution

It determines the number of skew braces with given multiplicative and additive groups of squarefree order, extending previous results on Hopf-Galois structures.

Findings

Count of skew braces for squarefree order n

Enumeration of skew braces with order as product of three primes

Connection between skew braces and Hopf-Galois structures

Abstract

Let $n \geq 1$ be a squarefree integer, and let $M$, $A$ be two groups of order $n$. Using our previous results on the enumeration of Hopf-Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group $M$ and additive group $A$. As an application, we enumerate skew braces whose order is the product of three distinct primes.