{
  "id": "1701.00141",
  "version": "v1",
  "published": "2016-12-31T16:44:01.000Z",
  "updated": "2016-12-31T16:44:01.000Z",
  "title": "The distinguishing number of groups based on the distinguishing number of subgroups",
  "authors": [
    "Saeid Alikhani",
    "Samaneh Soltani"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\Gamma$ be a group acting on a set $X$. The distinguishing number for this action of $\\Gamma$ on $X$, denoted by $D_{\\Gamma}(X)$, is the smallest natural number $k$ such that the elements of $X$ can be labeled with $k$ labels so that any label-preserving element of $\\Gamma$ fixes all $x \\in X$. In particular, if the action is faithful, then the only element of $\\Gamma$ preserving labels is the identity. In this paper, we obtain an upper bound on the distinguishing number of a set knowing the distinguishing number of a set under the action of a subgroup. By the concept of motion, we obtain an upper bound for the distinguishing number of a group. Motivated by a problem (Chan 2006), we characterize $D_{\\Gamma,H}(X)$ which is the smallest number of labels admitting a labeling of $X$ such that the only elements of $\\Gamma$ that induce label-preserving permutations lie in $H$. Finally, we state two algorithms for obtaining an upper and a lower bound for $D_{\\Gamma , H}(X)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-31T16:44:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15",
      "05C15"
    ],
    "keywords": [
      "distinguishing number",
      "upper bound",
      "induce label-preserving permutations lie",
      "smallest natural number",
      "smallest number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}