{
  "id": "1205.3533",
  "version": "v1",
  "published": "2012-05-16T00:27:22.000Z",
  "updated": "2012-05-16T00:27:22.000Z",
  "title": "Alternatives for pseudofinite groups",
  "authors": [
    "Abderezak Ould Houcine",
    "Françoise Point"
  ],
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "The famous Tits' alternative states that a linear group either contains a nonabelian free group or is soluble-by-(locally finite). We study in this paper similar alternatives in pseudofinite groups. We show for instance that an $\\aleph_{0}$-saturated pseudofinite group either contains a subsemigroup of rank $2$ or is nilpotent-by-(uniformly locally finite). We call a class of finite groups $G$ weakly of bounded rank if the radical $rad(G)$ has a bounded Pr\\\"ufer rank and the index of the sockel of $G/rad(G)$ is bounded. We show that an $\\aleph_{0}$-saturated pseudo-(finite weakly of bounded rank) group either contains a nonabelian free group or is nilpotent-by-abelian-by-(uniformly locally finite). We also obtain some relations between this kind of alternatives and amenability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-16T00:27:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonabelian free group",
      "uniformly locally finite",
      "bounded rank",
      "paper similar alternatives",
      "finite groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.3533O"
    }
  }
}