{
  "id": "1108.1784",
  "version": "v2",
  "published": "2011-08-08T19:24:51.000Z",
  "updated": "2012-02-25T00:26:31.000Z",
  "title": "The probability that a pair of elements of a finite group are conjugate",
  "authors": [
    "Simon R. Blackburn",
    "John R. Britnell",
    "Mark Wildon"
  ],
  "comment": "34 pages, corrected version, to appear in Journal of the London Mathematical Society",
  "doi": "10.1112/jlms/jds022",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "Let $G$ be a finite group, and let $\\kappa(G)$ be the probability that elements $g$, $h\\in G$ are conjugate, when $g$ and $h$ are chosen independently and uniformly at random. The paper classifies those groups $G$ such that $\\kappa(G) \\geq 1/4$, and shows that $G$ is abelian whenever $\\kappa(G)|G| < 7/4$. It is also shown that $\\kappa(G)|G|$ depends only on the isoclinism class of $G$. Specialising to the symmetric group $S_n$, the paper shows that $\\kappa(S_n) \\leq C/n^2$ for an explicitly determined constant $C$. This bound leads to an elementary proof of a result of Flajolet \\emph{et al}, that $\\kappa(S_n) \\sim A/n^2$ as $n\\rightarrow \\infty$ for some constant $A$. The same techniques provide analogous results for $\\rho(S_n)$, the probability that two elements of the symmetric group have conjugates that commute.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-02-25T00:26:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60",
      "20B30",
      "20E45",
      "05A05",
      "05A16"
    ],
    "keywords": [
      "finite group",
      "probability",
      "symmetric group",
      "elementary proof",
      "paper classifies"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.1784B"
    }
  }
}