{
  "id": "1704.02509",
  "version": "v1",
  "published": "2017-04-08T16:10:07.000Z",
  "updated": "2017-04-08T16:10:07.000Z",
  "title": "Characterizations of some classes of finite $σ$-soluble $Pσ T$-groups",
  "authors": [
    "Alexander N. Skiba"
  ],
  "comment": "15 pages. arXiv admin note: text overlap with arXiv:1611.06569",
  "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 $\\sigma$-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 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 exactly 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 ${\\sigma}$-permutable 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 characterizations of finite $\\sigma$-soluble groups $G$ in which $\\sigma$-permutability is a transitive relation in $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-08T16:10:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20D15",
      "20D30"
    ],
    "keywords": [
      "characterizations",
      "complete hall",
      "finite group",
      "chief factor",
      "soluble groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}