{
  "id": "1209.0954",
  "version": "v1",
  "published": "2012-09-05T12:53:05.000Z",
  "updated": "2012-09-05T12:53:05.000Z",
  "title": "Collineation group as a subgroup of the symmetric group",
  "authors": [
    "Fedor Bogomolov",
    "Marat Rovinsky"
  ],
  "comment": "9 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Let $\\Psi$ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension $\\ge 3$ over a field. Let $H$ be a closed (in the pointwise convergence topology) subgroup of the permutation group $\\mathfrak{S}_{\\Psi}$ of the set $\\Psi$. Suppose that $H$ contains the projective group and an arbitrary self-bijection of $\\Psi$ transforming a triple of collinear points to a non-collinear triple. It is well-known from \\cite{KantorMcDonough} that if $\\Psi$ is finite then $H$ contains the alternating subgroup $\\mathfrak{A}_{\\Psi}$ of $\\mathfrak{S}_{\\Psi}$. We show in Theorem \\ref{density} below that $H=\\mathfrak{S}_{\\Psi}$, if $\\Psi$ is infinite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-05T12:53:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric group",
      "collineation group",
      "one-dimensional vector subspaces",
      "collinear points",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.0954B"
    }
  }
}