{
  "id": "2309.09906",
  "version": "v1",
  "published": "2023-09-18T16:14:18.000Z",
  "updated": "2023-09-18T16:14:18.000Z",
  "title": "The Erdős-Ko-Rado Theorem for non-quasiprimitive groups of degree $3p$",
  "authors": [
    "Roghayeh Maleki",
    "Andriaherimanana Sarobidy Razafimahatratra"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "The \\emph{intersection density} of a finite transitive group $G\\leq \\operatorname{Sym}(\\Omega)$ is the rational number $\\rho(G)$ given by the ratio between the maximum size of a subset of $G$ in which any two permutations agree on some elements of $\\Omega$ and the order of a point stabilizer of $G$. In 2022, Meagher asked whether $\\rho(G)\\in \\{1,\\frac{3}{2},3\\}$ for any transitive group $G$ of degree $3p$, where $p\\geq 5$ is an odd prime. For the primitive case, it was proved in [\\emph{J. Combin. Ser. A}, 194:105707, 2023] that the intersection density is $1$. It is shown in this paper that the answer to this question is affirmative for non-quasiprimitive groups, unless possibly when $p = q+1$ is a Fermat prime and $\\Omega$ admits a unique $G$-invariant partition $\\mathcal{B}$ such that the induced action $\\overline{G}_\\mathcal{B}$ of $G$ on $\\mathcal{B}$ is an almost simple group containing $\\operatorname{PSL}_{2}(q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-18T16:14:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C69",
      "20B05"
    ],
    "keywords": [
      "non-quasiprimitive groups",
      "erdős-ko-rado theorem",
      "finite transitive group",
      "permutations agree",
      "point stabilizer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}