{
  "id": "1611.06569",
  "version": "v1",
  "published": "2016-11-20T19:00:27.000Z",
  "updated": "2016-11-20T19:00:27.000Z",
  "title": "On finite $PσT$-groups",
  "authors": [
    "Alexander N. Skiba"
  ],
  "comment": "11 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $\\sigma =\\{\\sigma_{i} | i\\in I\\}$ be some partition of the set of all primes $\\Bbb{P}$ and $G$ a finite group. $G$ is said to be \\emph{$\\sigma$-soluble} if every chief factor $H/K$ of $G$ is a $\\sigma_{i}$-group for some $i=i(H/K)$. A set ${\\cal H}$ of subgroups of $G$ is said to be a \\emph{complete Hall $\\sigma $-set} of $G$ if every member $\\ne 1$ of ${\\cal H}$ is a Hall $\\sigma_{i}$-subgroup of $G$ for some $\\sigma_{i}\\in \\sigma $ and ${\\cal H}$ contains exact one Hall $\\sigma_{i}$-subgroup of $G$ for every $i \\in I$ such that $\\sigma_{i}\\cap \\pi (G)\\ne \\emptyset$. A subgroup $A$ of $G$ is said to be \\emph{${\\sigma}$-permutable} or \\emph{${\\sigma}$-quasinormal} in $G$ if $G$ has a complete Hall $\\sigma$-set $\\cal H$ such that $AH^{x}=H^{x}A$ for all $x\\in G$ and all $H\\in \\cal H$. We obtain a characterization of finite $\\sigma$-soluble groups $G$ in which $\\sigma$-quasinormality is a transitive relation in $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-20T19:00:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20D15",
      "20D30"
    ],
    "keywords": [
      "finite group",
      "chief factor",
      "contains exact",
      "complete hall",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}