Determining skew left braces of size np

TL;DR

The authors classify skew left braces of size $np$ under conditions where every group of order $np$ has a normal subgroup of order $p$ and $p mid | ext{Aut}(E)|$ for all groups $E$ of order $n$, showing that such braces decompose as twofold semidirect products or their companions. They introduce the twofold semidirect product to connect additive and multiplicative structures via compatible homomorphisms $\sigma$ and $\tau$, and they exploit the regular-subgroup (Hol$(N)$) correspondence to reduce brace classification to group-theoretic data. An algorithm is provided to construct all braces of size $np$ from braces of size $n$ by analyzing orbits of Hom$(B_n,\mathbf Z_p^*)$ under $\operatorname{Aut}(B_n)$ and the induced actions on Hom$( (B_n,\circ),\mathbf Z_p^*)$, along with a Burnside-type fixed-point count. The method is applied to size $12p$ with $p\ge 7$, yielding explicit totals that depend on $p\bmod 12$ and resolving conjectures on the numbers of such braces, thereby enriching the understanding of brace extensions and their role in set-theoretic Yang–Baxter solutions. The results provide concrete, computation-backed classifications across all 5 groups of order $12$, and a complete enumeration for $12p$ demonstrates the practical impact of the twofold dsdp framework.

Abstract

We define the twofold semidirect product of two skew left braces, in which both the additive and multiplicative groups are semidirect products of the corresponding groups of the given skew left braces. We consider an odd prime $p$ and an integer $n$ satisfying $p\nmid n$, $p\nmid|\mathrm{Aut}(E)|$ for every group $E$ of order $n$ and such that each group of order $np$ has a unique $p$-Sylow subgroup. Under these conditions, we prove that any skew left brace of size $np$ is either a twofold semidirect product of the trivial brace of size $p$ and a skew left brace of size $n$ or a companion skew left brace of that one. We develop an algorithm to obtain all skew left braces of size $np$ from the skew left braces of size $n$ and provide a formula to count them. We use this result to describe all skew left braces of size $12p$ for $p\geq 7$, which proves a conjecture of V.G. Bardakov, M.V. Neshchadim and M.K. Yadav.

Theorems & Definitions (37)

• Proposition 2.1
• proof
• Definition 2.2
• Lemma 2.4
• Proposition 2.5: GV Theorem 4.2
• Lemma 2.6
• proof
• Remark 2.7
• Lemma 2.8
• proof