On $\Pi$-permutable subgroups of finite groups

Abstract

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $\Pi$ a non-empty 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 exact one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \Pi$ such that $\sigma_i\cap \pi(G)\neq\emptyset$. A subgroup $H$ of $G$ is called \emph{$\Pi$-quasinormal} or \emph{$\Pi$-permutable} in $G$ if $G$ possesses a complete Hall $\Pi$-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 $\Pi$-permutable in $G$. Some known results are generalized.