Finite groups whose $n$-maximal subgroups are $σ$-subnormal

Let $σ=\{σ_{i} | i\in I\}$ 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 $σ$-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$, for some $i\in I$, and $\cal H$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in σ(G)$. A subgroup $H$ of $G$ is said to be: \emph{$σ$-permutable} or \emph{$σ$-quasinormal} in $G$ if $G$ possesses a complete Hall $σ$-set set ${\cal H}$ such that $HA^{x}=A^{x}H$ for all $A\in {\cal H}$ and $x\in G$: \emph{$σ$-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 $σ_{i}$-group for some $σ_{i}\in σ$ for all $i=1, \ldots t$. If each $n$-maximal subgroup of $G$ is $σ$-subnormal ($σ$-quasinormal, respectively) in $G$ but, in the case $n > 1$, some $(n-1)$-maximal subgroup is not $σ$-subnormal (not $σ$-quasinormal, respectively)) in $G$, we write $m_σ(G)=n$ ($m_{σq}(G)=n$, respectively). In this paper, we show that the parameters $m_σ(G)$ and $m_{σq}(G)$ make possible to bound the $σ$-nilpotent length $\ l_σ(G)$ (see below the definitions of the terms employed), the rank $r(G)$ and the number $|π(G)|$ of all distinct primes dividing the order $|G|$ of a finite soluble group $G$. We also give conditions under which a finite group is $σ$-soluble or $σ$-nilpotent, and describe the structure of a finite soluble group $G$ in the case when $m_σ(G)=|π(G)|$. Some known results are generalized.

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