{
  "id": "2404.00004",
  "version": "v1",
  "published": "2024-02-18T20:56:43.000Z",
  "updated": "2024-02-18T20:56:43.000Z",
  "title": "A contribution to the theory of $σ$-properties of a finite group",
  "authors": [
    "A-Ming Liu",
    "Wenbin Guo",
    "Vasily G. Safonov",
    "Alexander N. Skiba"
  ],
  "comment": "19 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-18T20:56:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20D15",
      "20D30"
    ],
    "keywords": [
      "finite group",
      "finite soluble group",
      "properties",
      "contribution",
      "chief factor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}