{
  "id": "1704.02472",
  "version": "v1",
  "published": "2017-04-08T10:39:05.000Z",
  "updated": "2017-04-08T10:39:05.000Z",
  "title": "Difference bases in dihedral groups",
  "authors": [
    "Taras Banakh",
    "Volodymyr Gavrylkiv"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "A subset $B$ of a group $G$ is called a difference basis of $G$ if each element $g\\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\\in B$. The smallest cardinality $|B|$ of a difference basis $B\\subset G$ is called the difference size of $G$ and is denoted by $\\Delta[G]$. The fraction $\\eth[G]:=\\Delta[G]/{\\sqrt{|G|}}$ is called the difference characteristic of $G$. We prove that for every $n\\in\\mathbb N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\\sqrt{2}\\le\\eth[D_{2n}]\\leq\\frac{48}{\\sqrt{586}}\\approx1.983$. Moreover, if $n\\ge 2\\cdot 10^{15}$, then $\\eth[D_{2n}]<\\frac{4}{\\sqrt{6}}\\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\\le80$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-08T10:39:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B10",
      "05E15",
      "20D60"
    ],
    "keywords": [
      "dihedral group",
      "difference basis",
      "difference characteristic",
      "difference size"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}