{
  "id": "2410.19422",
  "version": "v1",
  "published": "2024-10-25T09:27:38.000Z",
  "updated": "2024-10-25T09:27:38.000Z",
  "title": "Flag-transitive point-primitive quasi-symmetric $2$-designs with block intersection numbers $0$ and $y\\leq10$",
  "authors": [
    "Jianbing Lu",
    "Yu Zhuang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we show that for a non-trivial quasi-symmetric $2$-design $\\mathcal{D}$ with two block intersection numbers $x=0$ and $2\\leq y\\leq10$, if $G\\leq \\mathrm{Aut}(\\mathcal{D})$ is flag-transitive and point-primitive, then $G$ is either of affine type or almost simple type. Moreover, we prove that the socle of $G$ cannot be an alternating group. If the socle of $G$ is a sporadic group, then $\\mathcal{D}$ and $G$ must be one of the following: $\\mathcal{D}$ is a $2$-$(12,6,5)$ design with block intersection numbers $0,3$ and $G=\\mathrm{M}_{11}$, or $\\mathcal{D}$ is a $2$-$(22,6,5)$ design with block intersection numbers $0,2$ and $G=\\mathrm{M}_{22}$ or $\\mathrm{M}_{22}:2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-25T09:27:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "block intersection numbers",
      "flag-transitive point-primitive quasi-symmetric",
      "sporadic group",
      "non-trivial quasi-symmetric",
      "affine type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}