{
  "id": "1411.0848",
  "version": "v1",
  "published": "2014-11-04T10:19:37.000Z",
  "updated": "2014-11-04T10:19:37.000Z",
  "title": "Commuting probabilities of finite groups",
  "authors": [
    "Sean Eberhard"
  ],
  "comment": "14 pages",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \\subset (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-04T10:19:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D60",
      "20F24",
      "15A63"
    ],
    "keywords": [
      "finite group",
      "commuting probability",
      "randomly chosen group elements commute",
      "abelian groups",
      "limit points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.0848E"
    }
  }
}