Determining skew left braces of size np

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 (Formula presented.), (Formula presented.) 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 (Formula presented.), which proves a conjecture of V.G. Bardakov, M.V. Neshchadim and M.K. Yadav.