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Finite groups whose $n$-maximal subgroups are $sigma$-subnormal | Wenbin Guo
; Alexander N. Skiba
; | Date: |
11 Aug 2016 | Abstract: | Let $sigma ={sigma_{i} | iin I}$ 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 | Services: | Forum | Review | PDF | Favorites |
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