{
  "id": "1205.2824",
  "version": "v6",
  "published": "2012-05-13T03:19:35.000Z",
  "updated": "2014-05-24T12:53:14.000Z",
  "title": "How many tuples of group elements have a given property?",
  "authors": [
    "Anton A. Klyachko",
    "Anna A. Mkrtchyan"
  ],
  "comment": "With an appendix by D. V. Trushin, 9 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V3: a reference to Mathoverflow is added. V4: Corollary 5 is strengthened; the construction of the matrix $A(\\phi)$ is clarified slightly. V5: The main theorem is strengthened; a corollary is added. V6: minor misprint correction",
  "journal": "International Journal of Algebra and Computation, 2014, 24:4, 413-428",
  "doi": "10.1142/S0218196714500192",
  "categories": [
    "math.GR",
    "math.LO"
  ],
  "abstract": "Generalising Solomon's theorem, C. Gordon and F. Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of i-th unknown in j-th equation is less than the number of unknowns. We generalise this result in two directions: first, we consider equations with coefficients, and secondly, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusing facts; for example, the number of group elements whose squares lie in a given subgroup is divisible by the order this subgroup.",
  "revisions": [
    {
      "version": "v6",
      "updated": "2014-05-24T12:53:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60",
      "20F70",
      "20F10"
    ],
    "keywords": [
      "group elements",
      "theorem implies",
      "group language",
      "first-order formulae",
      "squares lie"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2824K"
    }
  }
}