{
  "id": "1301.6361",
  "version": "v9",
  "published": "2013-01-27T14:46:12.000Z",
  "updated": "2014-11-04T13:54:00.000Z",
  "title": "On partial $Π$-property of subgroups of finite groups",
  "authors": [
    "Xiaoyu Chen",
    "Wenbin Guo"
  ],
  "comment": "This is a final corrected version of the version published in J. Group Theory!! arXiv admin note: text overlap with arXiv:1307.0089",
  "journal": "J. Group Theory 16 (2013), 745-766",
  "doi": "10.1515/jgt-2013-0018",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\\Pi$-property in $G$ if there exists a chief series $\\mathit{\\Gamma}_G:1=G_0<G_1<\\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\\leq i\\leq n$) of $\\mathit{\\Gamma}_G$, $|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1}\\cap G_i/G_{i-1})|$ is a $\\pi(HG_{i-1}/G_{i-1}\\cap G_i/G_{i-1})$-number. Our main results are listed here: Theorem A. Let $\\mathfrak{F}$ be a solubly saturated formation containing $\\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\\in \\mathfrak{F}$. Let $X\\unlhd G$ such that $F_p^*(E)\\leq X\\leq E$. Suppose that for any Sylow $p$-subgroup $P$ of $X$, every maximal subgroup of $P$ satisfies partial $\\Pi$-property in $G$. Then one of the following holds: (1) $G\\in \\mathfrak{G}_{p'}\\mathfrak{F}$. (2) $X/O_{p'}(X)$ is a quasisimple group with Sylow $p$-subgroups of order $p$. In particular, if $X=F_p^*(E)$, then $X/O_{p'}(X)$ is a simple group. Theorem B. Let $\\mathfrak{F}$ be a solubly saturated formation containing $\\mathfrak{U}$ and $E$ a normal subgroup of $G$ with $G/E\\in \\mathfrak{F}$. Suppose that for any Sylow $p$-subgroup $P$ of $F_p^*(E)$, every cyclic subgroup of $P$ of prime order or order 4 (when $P$ is not quaternion-free) satisfies partial $\\Pi$-property in $G$. Then $G\\in \\mathfrak{G}_{p'}\\mathfrak{F}$.",
  "revisions": [
    {
      "version": "v8",
      "updated": "2014-05-05T10:44:24.000Z",
      "comment": "This is a final corrected version of the version published in J. Group Theory!!"
    },
    {
      "version": "v9",
      "updated": "2014-11-04T13:54:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20D15",
      "20D20"
    ],
    "keywords": [
      "finite group",
      "satisfies partial",
      "normal subgroup",
      "solubly saturated formation containing",
      "chief series"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.6361C"
    }
  }
}