{
  "id": "math/0601414",
  "version": "v1",
  "published": "2006-01-17T14:28:41.000Z",
  "updated": "2006-01-17T14:28:41.000Z",
  "title": "The distinguishing number of the direct product and wreath product action",
  "authors": [
    "Melody Chan"
  ],
  "comment": "16 pages, to appear in the Journal of Algebraic Combinatorics",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product of G and H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S_m x S_n on [m] x [n].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-01-17T14:28:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "20B25",
      "20D60"
    ],
    "keywords": [
      "distinguishing number",
      "wreath product action",
      "direct product",
      "nontrivial group element induces",
      "important product actions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1414C"
    }
  }
}