On $\Pi$-permutable subgroups of finite groups

Abstract

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and $\Pi$ a non-empty subset of the set $\sigma$. A set $\mathcal{H}$ of subgroups of a finite group $G$ is said to be a \emph{ complete Hall $\Pi$-set} of $G$ if every member of $\mathcal{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \Pi$ and $\mathcal{H}$ contains exact one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \Pi$ such that ${\sigma }_i\cap \pi (G)\ne \varnothing$. A subgroup $H$ of $G$ is called \emph{$\Pi$-quasinormal} or \emph{$\Pi$-permutable} in $G$ if $G$ possesses a complete Hall $\Pi$-set $\mathcal{H}=\{H_1,\dots ,H_t\}$ such that $A{H}_i^x=H_i^{x}A$ 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.

11 pages, conference