{
  "id": "1601.02135",
  "version": "v1",
  "published": "2016-01-09T17:27:35.000Z",
  "updated": "2016-01-09T17:27:35.000Z",
  "title": "Ascending chains of finitely generated subgroups",
  "authors": [
    "Mark Shusterman"
  ],
  "categories": [
    "math.GR",
    "math.AC"
  ],
  "abstract": "We show that a nonempty family of $n$-generated subgroups of a pro-$p$ group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-$p$ groups. To demonstrate this, we show that in various pro-$p$ groups $\\Gamma$ (e.g. free pro-$p$ groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup $H \\neq 1$ is the greatest subgroup of $\\Gamma$ containing $H$ as an open subgroup. We also show that an ascending sequence of $n$-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich-Myasnikov).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-09T17:27:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20E18",
      "20E26",
      "20E36",
      "20F65",
      "20F05"
    ],
    "keywords": [
      "finitely generated subgroup",
      "ascending chains",
      "noetherian induction",
      "nonsolvable demushkin groups",
      "maximal element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102135S"
    }
  }
}