{
  "id": "1710.05378",
  "version": "v1",
  "published": "2017-10-15T18:51:17.000Z",
  "updated": "2017-10-15T18:51:17.000Z",
  "title": "On generalized $σ$-soluble groups",
  "authors": [
    "Jianhong Huang",
    "Bin Hu",
    "Alexander N. Skiba"
  ],
  "comment": "13 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $\\sigma =\\{\\sigma_{i} | i\\in I\\}$ be a partition of the set of all primes $\\Bbb{P}$ and $G$ a finite group. Let $\\sigma (G)=\\{\\sigma _{i} : \\sigma _{i}\\cap \\pi (G)\\ne \\emptyset$. 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 $i\\in I$ and $\\cal H$ contains exactly one Hall $\\sigma _{i}$-subgroup of $G$ for every $i$ such that $\\sigma _{i}\\in \\sigma (G)$. We say that $G$ is $\\sigma$-full if $G$ possesses a complete Hall $\\sigma $-set. A complete Hall $\\sigma $-set $\\cal H$ of $G$ is said to be a $\\sigma$-basis of $G$ if every two subgroups $A, B \\in\\cal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $\\sigma$-basis. In particular, we prove that if $G$ has a a $\\sigma$-basis, then $G$ is generalized $\\sigma$-soluble, that is, $G$ has a complete Hall $\\sigma $-set and for every chief factor $H/K$ of $G$ we have $|\\sigma (H/K)|\\leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $\\sigma$-full. Then every complete Hall $\\sigma$-set of $G$ forms a $\\sigma$-basis of $G$ if and only if $G$ is generalized $\\sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $\\sigma (H/K)=\\sigma (G/C_{G}(H/K))$ or $\\sigma (H/K) =\\{\\sigma _{i}\\}$ and $G/C_{G}(H/K)$ is a $\\sigma _{i} \\cup \\sigma _{j}$-group for some $i\\ne j$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-15T18:51:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20D10",
      "20D15"
    ],
    "keywords": [
      "complete hall",
      "finite group",
      "chief factor",
      "automorphism group",
      "finite partially soluble groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}