{
  "id": "1608.03353",
  "version": "v1",
  "published": "2016-08-11T02:57:59.000Z",
  "updated": "2016-08-11T02:57:59.000Z",
  "title": "Finite groups whose $n$-maximal subgroups are $σ$-subnormal",
  "authors": [
    "Wenbin Guo",
    "Alexander N. Skiba"
  ],
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $\\sigma =\\{\\sigma_{i} | i\\in I\\}$ be some partition of the set of all primes $\\Bbb{P}$. 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 $i\\in I$, and $\\cal H$ contains exact one Hall $\\sigma_{i}$-subgroup of $G$ for every $\\sigma_{i}\\in \\sigma (G)$. A subgroup $H$ of $G$ is said to be: \\emph{$\\sigma$-permutable} or \\emph{$\\sigma$-quasinormal} in $G$ if $G$ possesses a complete Hall $\\sigma$-set set ${\\cal H}$ such that $HA^{x}=A^{x}H$ for all $A\\in {\\cal H}$ and $x\\in G$: \\emph{${\\sigma}$-subnormal} in $G$ if there is a subgroup chain $A=A_{0} \\leq A_{1} \\leq \\cdots \\leq A_{t}=G$ such that either $A_{i-1}\\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\\sigma_{i}$-group for some $\\sigma_{i}\\in \\sigma$ for all $i=1, \\ldots t$. If each $n$-maximal subgroup of $G$ is $\\sigma$-subnormal ($\\sigma$-quasinormal, respectively) in $G$ but, in the case $ n > 1$, some $(n-1)$-maximal subgroup is not $\\sigma$-subnormal (not $\\sigma$-quasinormal, respectively)) in $G$, we write $m_{\\sigma}(G)=n$ ($m_{\\sigma q}(G)=n$, respectively). In this paper, we show that the parameters $m_{\\sigma}(G)$ and $m_{\\sigma q}(G)$ make possible to bound the $\\sigma$-nilpotent length $ \\ l_{\\sigma}(G)$ (see below the definitions of the terms employed), the rank $r(G)$ and the number $|\\pi (G)|$ of all distinct primes dividing the order $|G|$ of a finite soluble group $G$. We also give conditions under which a finite group is $\\sigma$-soluble or $\\sigma$-nilpotent, and describe the structure of a finite soluble group $G$ in the case when $m_{\\sigma}(G)=|\\pi (G)|$. Some known results are generalized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-11T02:57:59.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximal subgroup",
      "finite group",
      "finite soluble group",
      "quasinormal",
      "nilpotent length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}