{
  "id": "2107.14208",
  "version": "v1",
  "published": "2021-07-29T17:47:10.000Z",
  "updated": "2021-07-29T17:47:10.000Z",
  "title": "On relational complexity and base size of finite primitive groups",
  "authors": [
    "Veronica Kelsey",
    "Colva M. Roney-Dougal"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "In this paper we show that if $G$ is a primitive subgroup of $S_{n}$ that is not large base, then any irredundant base for $G$ has size at most $5 \\log n$. This is the first logarithmic bound on the size of an irredundant base for such groups, and is best possible up to a small constant. As a corollary, the relational complexity of $G$ is at most $5 \\log n+1$, and the maximal size of a minimal base and the height are both at most $5 \\log n.$ Furthermore, we deduce that a base for $G$ of size at most $5 \\log n$ can be computed in polynomial time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-29T17:47:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B15",
      "20B25",
      "20E32",
      "20-08"
    ],
    "keywords": [
      "finite primitive groups",
      "relational complexity",
      "base size",
      "irredundant base",
      "first logarithmic bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}