{
  "id": "2311.04057",
  "version": "v1",
  "published": "2023-11-07T15:08:00.000Z",
  "updated": "2023-11-07T15:08:00.000Z",
  "title": "Finite permutation groups of rank $3$",
  "authors": [
    "Hong Yi Huang",
    "Cai Heng Li",
    "Yan Zhou Zhu"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-07T15:08:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite permutation groups",
      "unique non-trivial block system",
      "finite primitive permutation groups",
      "main theorem divides",
      "regular normal subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}