Finite groups whose $n$-maximal subgroups are $\sigma$-subnormal

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.