Finite groups whose n-maximal subgroups are σ-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 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 exactly one Hall $\sigma~_{i}$-subgroup of $G$ for every $\sigma~_{i}\in~\sigma~(G)$. A subgroup $H$ of $G$ is said to be: $\sigma$-permutable or $\sigma$-quasinormal in $G$ if $G$ possesses a complete Hall $\sigma$-set ${\cal~H}$ such that $HA^{x}=A^{x}H$ for all $A\in~{\cal~H}$ and $x\in~G$: ${\sigma}$-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 $M_n~ $n$-maximal subgroup of $G$. 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 the 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.