{
  "id": "2402.01072",
  "version": "v2",
  "published": "2023-12-26T09:45:39.000Z",
  "updated": "2024-02-06T02:42:08.000Z",
  "title": "The influence of weakly $SΦ$-supplemented subgroups on fusion systems of finite groups",
  "authors": [
    "Shengmin Zhang",
    "Zhencai Shen"
  ],
  "comment": "This paper is not complete",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a finite group and $H$ be a subgroup of $G$. Then $H$ is called a weakly $S\\Phi$-supplemented subgroup of $G$, if there exists a subgroup $T$ of $G$ such that $G =HT$ and $H \\cap T \\leq \\Phi (H) H_{sG}$, where $H_{sG}$ denotes the subgroup of $H$ generated by all subgroups of $H$ which are $S$-permutable in $G$. Let $p$ be a prime, $S$ be a $p$-group and $\\mathcal{F}$ be a saturated fusion system over $S$. Then $\\mathcal{F}$ is said to be supersolvable, if there exists a series of $S$, namely $1 = S_0 \\leq S_1 \\leq \\cdots \\leq S_n = S$, such that $S_{i+1}/S_i$ is cyclic, $i=0,1,\\cdots, n-1$, $S_i$ is strongly $\\mathcal{F}$-closed, $i=0,1,\\cdots,n$. In this paper, we investigate the structure of fusion system $\\mathcal{F}_S (G)$ under the assumption that certain subgroups of $S$ are weakly $S\\Phi$-supplemented in $G$, and obtain several new characterizations of supersolvability of $\\mathcal{F}_S (G)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-02-06T02:42:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20D15",
      "20D20"
    ],
    "keywords": [
      "finite group",
      "supplemented subgroup",
      "saturated fusion system",
      "assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}