{
  "id": "1702.02631",
  "version": "v1",
  "published": "2017-02-08T21:59:16.000Z",
  "updated": "2017-02-08T21:59:16.000Z",
  "title": "Self-linked sets in finite groups",
  "authors": [
    "Taras Banakh",
    "Volodymyr Gavrylkiv"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "A subset $A$ of a group $G$ is called self-linked if $A\\cap gA\\ne\\emptyset$ for every $g\\in G$. The smallest cardinality $|A|$ of a self-linked subset $A\\subset G$ is called the self-linked number $sl(G)$ of $G$. In the paper we find lower and upper bounds for the self-linked number $sl(G)$ of a finite group $G$ and prove that $$\\frac{1+\\sqrt{4|G|+4|G_2|-7}}2\\le sl(G)\\le \\frac{\\sqrt[4]{|G|\\ln(|G|-1)}+\\sqrt{\\ln 4}}{\\sqrt[4]{|G|\\ln(|G|-1)}-\\sqrt{\\ln 4}}\\cdot \\sqrt{|G|\\ln(|G|-1)}$$where $G_2=\\{g\\in G:g^{-1}=g\\}$ is the set of elements of order at most two in $G$. Also we calculate the self-linked numbers of all Abelian groups of cardinality $\\le95$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-08T21:59:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E15"
    ],
    "keywords": [
      "finite group",
      "self-linked sets",
      "self-linked number",
      "abelian groups",
      "smallest cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}