On $Π$-permutable subgroups of finite groups

Abstract

Let $σ=\{σ_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $Π$ a non-empty subset of the set $σ$. A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{ complete Hall $Π$-set} of $G$ if every member of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $σ_{i}\in Π$ and ${\cal H}$ contains exact one Hall $σ_{i}$-subgroup of $G$ for every $σ_{i}\in Π$ such that $σ_i\cap π(G)\neq\emptyset$. A subgroup $H$ of $G$ is called \emph{$Π$-quasinormal} or \emph{$Π$-permutable} in $G$ if $G$ possesses a complete Hall $Π$-set ${\cal H}=\{H_{1}, \ldots , H_{t} \}$ such that $AH_{i}^{x}=H_{i}^{x}A$ for any $i$ and all $x\in G$. We study the embedding properties of $H$ under the hypothesis that $H$ is $Π$-permutable in $G$. Some known results are generalized.