{
  "id": "1805.05097",
  "version": "v1",
  "published": "2018-05-14T10:13:40.000Z",
  "updated": "2018-05-14T10:13:40.000Z",
  "title": "On an open problem of Skiba",
  "authors": [
    "Zhenfeng Wu",
    "Chi Zhang",
    "Wenbin Guo"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $\\sigma=\\{\\sigma_{i}|i\\in I\\}$ be some partition of the set $\\mathbb{P}$ of all primes, that is, $\\mathbb{P}=\\bigcup_{i\\in I}\\sigma_{i}$ and $\\sigma_{i}\\cap \\sigma_{j}=\\emptyset$ for all $i\\neq j$. Let $G$ be a finite group. A set $\\mathcal {H}$ of subgroups of $G$ is said to be a complete Hall $\\sigma$-set of $G$ if every non-identity member of $\\mathcal {H}$ is a Hall $\\sigma_{i}$-subgroup of $G$ and $\\mathcal {H}$ contains exactly one Hall $\\sigma_{i}$-subgroup of $G$ for every $\\sigma_{i}\\in \\sigma(G)$. $G$ is said to be a $\\sigma$-group if it possesses a complete Hall $\\sigma$-set. A $\\sigma$-group $G$ is said to be $\\sigma$-dispersive provided $G$ has a normal series $1 = G_1<G_2<\\cdots< G_t< G_{t+1} = G$ and a complete Hall $\\sigma$-set $\\{H_{1}, H_{2}, \\cdots, H_{t}\\}$ such that $G_iH_i = G_{i+1}$ for all $i= 1,2,\\ldots t$. In this paper, we give a characterizations of $\\sigma$-dispersive group, which give a positive answer to an open problem of Skiba in the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-05-14T10:13:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "open problem",
      "complete hall",
      "finite group",
      "non-identity member",
      "normal series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}