{
  "id": "2202.09705",
  "version": "v2",
  "published": "2022-02-20T00:55:52.000Z",
  "updated": "2022-02-25T03:27:19.000Z",
  "title": "The minimal size of a generating set for primitive $\\frac{3}{2}$-transitive groups",
  "authors": [
    "Dmitry Churikov",
    "Andrey V. Vasil'ev",
    "Maria A. Zvezdina"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "We refer to $d(G)$ as the minimal cardinality of a generating set of a finite group $G$, and say that $G$ is $d$-generated if $d(G)\\leq d$. A transitive permutation group $G$ is called $\\frac{3}{2}$-transitive if a point stabilizer $G_\\alpha$ is nontrivial and its orbits distinct from $\\{\\alpha\\}$ are of the same size. We prove that $d(G)\\leq4$ for every primitive $\\frac{3}{2}$-transitive permutation group $G$, moreover, $G$ is $2$-generated except for the very particular solvable affine groups that we completely describe. In particular, all finite $2$-transitive and $2$-homogeneous groups are $2$-generated. We also show that every finite group whose abelian subgroups are cyclic is $2$-generated, and so is every Frobenius complement.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-02-25T03:27:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B05",
      "20B15",
      "20B20"
    ],
    "keywords": [
      "generating set",
      "transitive groups",
      "minimal size",
      "transitive permutation group",
      "finite group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}