{
  "id": "1606.03197",
  "version": "v1",
  "published": "2016-06-10T06:08:57.000Z",
  "updated": "2016-06-10T06:08:57.000Z",
  "title": "On $Π$-permutable subgroups of finite groups",
  "authors": [
    "Wenbin Guo",
    "A. N. Skiba"
  ],
  "comment": "11 pages, conference",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $\\sigma =\\{\\sigma_{i} | i\\in I\\}$ be some partition of the set of all primes $\\Bbb{P}$ and $\\Pi$ a non-empty subset of the set $\\sigma$. A set ${\\cal H}$ of subgroups of a finite group $G$ is said to be a \\emph{ complete Hall $\\Pi $-set} of $G$ if every member of ${\\cal H}$ is a Hall $\\sigma_{i}$-subgroup of $G$ for some $\\sigma_{i}\\in \\Pi$ and ${\\cal H}$ contains exact one Hall $\\sigma_{i}$-subgroup of $G$ for every $\\sigma_{i}\\in \\Pi$ such that $\\sigma_i\\cap \\pi(G)\\neq\\emptyset$. A subgroup $H$ of $G$ is called \\emph{$\\Pi$-quasinormal} or \\emph{$\\Pi$-permutable} in $G$ if $G$ possesses a complete Hall $\\Pi$-set ${\\cal H}=\\{H_{1}, \\ldots , H_{t} \\}$ such that $AH_{i}^{x}=H_{i}^{x}A$ for any $i$ and all $x\\in G$. We study the embedding properties of $H$ under the hypothesis that $H$ is $\\Pi$-permutable in $G$. Some known results are generalized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-10T06:08:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite group",
      "permutable subgroups",
      "complete hall",
      "contains exact",
      "non-empty subset"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}