{
  "id": "2009.09297",
  "version": "v1",
  "published": "2020-09-19T20:54:29.000Z",
  "updated": "2020-09-19T20:54:29.000Z",
  "title": "Frattini-injectivity and Maximal pro-$p$ Galois groups",
  "authors": [
    "Ilir Snopce",
    "Slobodan Tanushevski"
  ],
  "comment": "33 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We call a pro-$p$ group $G$ Frattini-injective if distinct finitely generated subgroups of $G$ have distinct Frattinis. This paper is an initial effort toward a systematic study of Frattini-injective pro-$p$ groups (and several other related concepts). Most notably, we classify the $p$-adic analytic and the solvable Frattini-injective pro-$p$ groups, and we describe the lattice of normal abelian subgroups of a Frattini-injective pro-$p$ group. We prove that every maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\\sqrt{-1}$ if $p=2$) is Frattini-injective. In addition, we show that many substantial results on maximal pro-$p$ Galois groups are in fact consequences of Frattini-injectivity. For instance, a $p$-adic analytic or solvable pro-$p$ group is Frattini-injective if and only if it can be realized as a maximal pro-$p$ Galois group of a field that contains a primitive $p$th root of unity (and also contains $\\sqrt{-1}$ if $p=2$); and every Frattini-injective pro-$p$ group contains a unique maximal abelian normal subgroup.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-19T20:54:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "12F10",
      "12G05",
      "22E20"
    ],
    "keywords": [
      "galois group",
      "unique maximal abelian normal subgroup",
      "frattini-injective",
      "frattini-injectivity",
      "th root"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}