Double semidirect products and skew left braces of size np

Teresa Crespo, Daniel Gil-Muñoz, Anna Rio, Montserrat Vela

Published 2023-02-25Version 1

We define the double semidirect product of skew left braces and prove that if $p$ is an odd prime and $n$ is an integer such that $p\nmid n$ and each group of order $np$ has a unique $p$-Sylow subgroup, then any skew left brace of size $np$ is a double semidirect product of the trivial brace of size $p$ and a skew brace of size $n$. We develop an algorithm to obtain all braces of size $np$ from the set of braces of size $n$ and provide a formula to count them. We use the result to describe all braces of size $12p$ for $p\ge 7$ and prove a conjecture of Bardakov, M.V. Neshchadim and M.K. Yadav.