{
  "id": "2206.07863",
  "version": "v1",
  "published": "2022-06-16T00:38:30.000Z",
  "updated": "2022-06-16T00:38:30.000Z",
  "title": "Frattini-resistant direct products of pro-$p$ groups",
  "authors": [
    "Ilir Snopce",
    "Slobodan Tanushevski"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "A pro-$p$ group $G$ is called strongly Frattini-resistant if the function $H \\mapsto \\Phi(H)$, from the poset of all closed subgroups of $G$ into itself, is a poset embedding. Frattini-resistant pro-$p$ groups appear naturally in Galois theory. Indeed, every maximal pro-$p$ Galois group over a field that contains a primitive $p$th root of unity (and also contains $\\sqrt{-1}$ if $p=2$) is strongly Frattini-resistant. Let $G_1$ and $G_2$ be non-trivial pro-$p$ groups. We prove that $G_1 \\times G_2$ is strongly Frattini-resistant if and only if one of the direct factors $G_1$ or $G_2$ is torsion-free abelian and the other one has the property that all of its closed subgroups have torsion-free abelianization. As a corollary we obtain a group theoretic proof of a result of Koenigsmann on maximal pro-$p$ Galois groups that admit a non-trivial decomposition as a direct product. In addition, we give an example of a group that is not strongly Frattini-resistant, but has the property that its Frattini-function defines an order self-embedding of the poset of all topologically finitely generated subgroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-16T00:38:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "12F10",
      "22E20"
    ],
    "keywords": [
      "frattini-resistant direct products",
      "strongly frattini-resistant",
      "galois group",
      "closed subgroups",
      "group theoretic proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}