{
  "id": "2302.13098",
  "version": "v1",
  "published": "2023-02-25T15:06:31.000Z",
  "updated": "2023-02-25T15:06:31.000Z",
  "title": "Double semidirect products and skew left braces of size np",
  "authors": [
    "Teresa Crespo",
    "Daniel Gil-Muñoz",
    "Anna Rio",
    "Montserrat Vela"
  ],
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-25T15:06:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "skew left brace",
      "double semidirect product",
      "size np",
      "sylow subgroup",
      "odd prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}