{
  "id": "2403.17291",
  "version": "v1",
  "published": "2024-03-26T00:40:38.000Z",
  "updated": "2024-03-26T00:40:38.000Z",
  "title": "Probabilistic Generation of Finite Almost Simple Groups",
  "authors": [
    "Jason Fulman",
    "Daniele Garzoni",
    "Robert M. Guralnick"
  ],
  "comment": "26 pages",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "We prove that if G is a sufficiently large finite almost simple group of Lie type, then given a fixed nontrivial element x in G and a coset of G modulo its socle, the probability that x and a random element of the coset generate a subgroup containing the socle is uniformly bounded away from 0 (and goes to 1 if the field size goes to infinity). This is new even if G is simple. Together with results of Lucchini and Burness--Guralnick--Harper, this proves a conjecture of Lucchini and has an application to profinite groups. A key step in the proof is the determination of the limits for the proportion of elements in a classical group which fix no subspace of any bounded dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-26T00:40:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "simple group",
      "probabilistic generation",
      "coset generate",
      "profinite groups",
      "sufficiently large finite"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}