{
  "id": "math/0401305",
  "version": "v2",
  "published": "2004-01-22T19:20:06.000Z",
  "updated": "2005-05-27T22:27:22.000Z",
  "title": "Closed subgroups of the infinite symmetric group",
  "authors": [
    "George M. Bergman",
    "Saharon Shelah"
  ],
  "comment": "33 pages. See also http://math.berkeley.edu/~gbergman/papers and http://shelah.logic.at (pub. 823). To appear, Alg.Univ., issue honoring W.Taylor. Main results as before (greater length due to AU formatting), but some new results in \\S\\S11-12. Errors in subscripts between displays (12) and (13) fixed. Error in title of orig. posting fixed. 1 ref. added",
  "journal": "Algebra Universalis 55 (2006) 137-173",
  "doi": "10.1007/s00012-006-1959-z",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let S=Sym(\\Omega) be the group of all permutations of a countably infinite set \\Omega, and for subgroups G_1, G_2\\leq S let us write G_1\\approx G_2 if there exists a finite set U\\subseteq S such that < G_1\\cup U > = < G_2\\cup U >. It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Which of these classes a closed subgroup G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(\\Gamma)} of finite subsets \\Gamma\\subseteq\\Omega holds: (i) For every finite set \\Gamma, the subgroup G_{(\\Gamma)} has at least one infinite orbit in \\Omega. (ii) There exist finite sets \\Gamma such that all orbits of G_{(\\Gamma)} are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets \\Gamma such that the cardinalities of the orbits of G_{(\\Gamma)} have a common finite bound, but none such that G_{(\\Gamma)}=\\{1\\}. (iv) There exist finite sets \\Gamma such that G_{(\\Gamma)}=\\{1\\}. Some questions for further investigation are discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-05-27T22:27:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B07",
      "22F50"
    ],
    "keywords": [
      "infinite symmetric group",
      "finite set",
      "closed subgroup",
      "common finite bound",
      "cardinalities"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1305B"
    }
  }
}