On a family of simple skew braces

TL;DR

This paper introduces an infinite family of simple skew braces that are not traditional braces and are not derived from nonabelian simple groups, expanding the known examples in algebraic structures.

Contribution

It provides the first explicit construction of an infinite family of simple skew braces not arising from nonabelian simple groups.

Findings

For primes p, q with q dividing (p^p - 1)/(p - 1)

Exactly two simple skew braces of order p^p q (up to isomorphism)

First known examples of such skew braces not being braces or from nonabelian simple groups

Abstract

Several constructions have been given for families of simple braces, but few examples are known of simple skew braces which are not braces. In this paper, we exhibit the first example of an infinite family of simple skew braces which are not braces and which do not arise from nonabelian simple groups. More precisely, we show that, for any primes $p$, $q$ such that $q$ divides ${(p^p-1)}/{(p-1)}$, there are exactly two simple skew braces (up to isomorphism) of order $p^p q$.