On a family of simple skew braces
Nigel P. Byott
TL;DR
This work resolves the existence and full classification of an infinite family of simple skew braces not arising from nonabelian simple groups. For primes $p,q$ with $q\mid\frac{p^p-1}{p-1}$ and $n=p^p q$, the authors construct exactly two simple skew braces of order $n$, which are mutual opposites, via a regular subgroup in the holomorph of a carefully chosen additive group $N$ with $|N|=n$. The additive group is a semidirect product $N\cong P\rtimes Q$ with $P\cong C_p^p$ and $Q\cong C_q$, while the multiplicative group is a semidirect product $G\cong Q\rtimes P$ determined by a matrix of order $q$ in $\mathrm{GL}_p(\mathbb{F}_p)$; the opposite brace arises from a corresponding $G^{*}$. The paper proves $B$ and $B^{\mathrm{op}}$ are not isomorphic and shows $\mathrm{Aut}_{\mathrm{sb}}(B)$ is cyclic of order $p$, then establishes a complete classification: any simple skew brace of order $n$ is isomorphic to $B$ or $B^{\mathrm{op}}$, advancing the understanding of simple skew braces beyond those arising from nonabelian simple groups and linking to Hopf-Galois theory and the set-theoretic Yang-Baxter equation.
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 $p^p q$.