On $\Pi$-quasinormal 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 complete Hall $\Pi$ -set of G if every member $\ne 1$ of $\mathcal{H}$ is a Hall $\sigma _{i}$-subgroup of G for some $\sigma _{i}\in \Pi$ and $\mathcal{H}$ contains exactly one Hall $\sigma _{i}$-subgroup of G for every $\sigma _{i}\in \Pi$ such that $\sigma _i\cap \pi (G)\ne \emptyset$. A subgroup H of G is called $\Pi$-permutable or $\Pi$-quasinormal in G if G possesses a complete Hall $\Pi$-set $\mathcal{H}$ such that $AH^{x}=H^{x}A$ for all $H\in \mathcal{H}$ and $x\in G$. We study the embedding properties of H under the hypothesis that H is $\Pi$-permutable in G. Some well-known results are generalized.