On $\Pi$-permutable subgroups in finite groups

Abstract

UDC 512.542

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $\Pi$ be a nonempty subset of the set $\sigma.$  A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{complete Hall $\Pi$-set} of $G$ if every member of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $\sigma _{i}\in \Pi$ and ${\cal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma _{i}\in \Pi$ such that $\sigma_i\cap\pi(G)\neq\varnothing.$ A subgroup $A$ of $G$ is called (i) {${\cal H}^{G}$-permutable} if $AH^{x}=H^{x}A$ for $H\in {\cal H}$ and $x\in G$; (ii) {$\Pi$-permutable in $G$} if $A$ is ${\cal H}^{G}$-permutable for some complete Hall $\Pi$-set $\cal H$ of $G.$ 

We study the influence of $\Pi$-permutable subgroups on the structure of $G.$ In particular, we prove that if $\pi= \displaystyle\bigcup\nolimits_{\sigma_{i}\in \Pi} \sigma_{i}$ and $G =AB,$ where $A$ and $B$ are ${\cal H}^{G}$-permutable $\pi$-separable (respectively, $\pi$-closed) subgroups of $G,$ then $G$ is also $\pi$-separable (respectively, $\pi$-closed).  Some known results are generalized.