{
  "id": "2210.09635",
  "version": "v1",
  "published": "2022-10-18T07:08:51.000Z",
  "updated": "2022-10-18T07:08:51.000Z",
  "title": "The spread of finite and infinite groups",
  "authors": [
    "Scott Harper"
  ],
  "comment": "38 pages; survey article based on my lecture at Groups St Andrews 2022",
  "categories": [
    "math.GR"
  ],
  "abstract": "It is well known that every finite simple group has a generating pair. Moreover, Guralnick and Kantor proved that every finite simple group has the stronger property, known as $\\frac{3}{2}$-generation, that every nontrivial element is contained in a generating pair. Much more recently, this result has been generalised in three different directions, which form the basis of this survey article. First, we look at some stronger forms of $\\frac{3}{2}$-generation that the finite simple groups satisfy, which are described in terms of spread and uniform domination. Next, we discuss the recent classification of the finite $\\frac{3}{2}$-generated groups. Finally, we turn our attention to infinite groups, focusing on the recent discovery that the finitely presented simple groups of Thompson are also $\\frac{3}{2}$-generated, as are many of their generalisations. Throughout the article we pose open questions in this area, and we highlight connections with other areas of group theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-18T07:08:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite groups",
      "finite simple groups satisfy",
      "generating pair",
      "pose open questions",
      "stronger property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}