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28 March 2024
 
  » arxiv » 1608.3353

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Finite groups whose $n$-maximal subgroups are $sigma$-subnormal
Wenbin Guo ; Alexander N. Skiba ;
Date 11 Aug 2016
AbstractLet $sigma ={sigma_{i} | iin I}$ 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 $ e 1$ of ${cal H}$ is a Hall $sigma_{i}$-subgroup of $G$, for some $iin I$, 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 $Ain {cal H}$ and $xin G$: 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 $A_{i-1} rianglelefteq A_{i}$ 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.
Source arXiv, 1608.3353
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