{
  "id": "1707.07293",
  "version": "v1",
  "published": "2017-07-23T13:27:47.000Z",
  "updated": "2017-07-23T13:27:47.000Z",
  "title": "On the number of cyclic subgroups of a finite group",
  "authors": [
    "Igor Lima",
    "Martino Garonzi"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group and let $c(G)$ be the number of cyclic subgroups of $G$. We study the function $\\alpha(G) = c(G)/|G|$. We explore its basic properties and we point out a connection with the probability of commutation. For many families $\\mathscr{F}$ of groups we characterize the groups $G \\in \\mathscr{F}$ for which $\\alpha(G)$ is maximal and we classify the groups $G$ for which $\\alpha(G) > 3/4$. We also study the number of cyclic subgroups of a direct power of a given group deducing an asymptotic result and we characterize the equality $\\alpha(G) = \\alpha(G/N)$ when $G/N$ is a symmetric group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-23T13:27:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "cyclic subgroups",
      "finite group",
      "asymptotic result",
      "direct power",
      "basic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}