{
  "id": "1211.2559",
  "version": "v3",
  "published": "2012-11-12T10:51:24.000Z",
  "updated": "2013-01-29T10:30:07.000Z",
  "title": "Normal coverings and pairwise generation of finite alternating and symmetric groups",
  "authors": [
    "Daniela Bubboloni",
    "Cheryl E. Praeger",
    "Pablo Spiga"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "The normal covering number $\\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$ such that, for $G$ a symmetric group $\\Sym(n)$ or an alternating group $\\Alt(n)$, $\\gamma(G)\\geq cn$. This improves results of the first two authors who had earlier proved that $a\\varphi(n)\\leq\\gamma(G)\\leq 2n/3,$ for some positive constant $a$, where $\\varphi$ is the Euler totient function. Bounds are also obtained for the maximum size $\\kappa(G)$ of a set $X$ of conjugacy classes of $G=\\Sym(n)$ or $\\Alt(n)$ such that any pair of elements from distinct classes in $X$ generates $G$, namely $cn\\leq \\kappa(G)\\leq 2n/3$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-01-29T10:30:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B30"
    ],
    "keywords": [
      "symmetric group",
      "pairwise generation",
      "finite alternating",
      "euler totient function",
      "positive constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.2559B"
    }
  }
}