On n-${\sigma}$-embedded subgroups of finite groups

Abstract

Let ${\sigma =\{\sigma_i |i\in I\}}$ be some partition of the set of all primes ${\mathbb{P}}$, G be a finite group and ${\sigma(G)=\{\sigma_i|\sigma_i\cap \pi(G)\neq \emptyset\}}$. A set ${\mathcal{H}}$ of subgroups of G is said to be a complete Hall ${\sigma}$-set of G if every non-identity member of ${\mathcal{H}}$ is a Hall ${\sigma_i}$-subgroup of ${G}$ and ${\mathcal{H}}$ contains exactly one Hall ${\sigma_i}$-subgroup of G for every ${\sigma_i\in \sigma(G)}$. A subgroup H of G is ${\sigma}$-permutable in G if G possesses a complete Hall ${\sigma}$-set ${\mathcal{H}}$ such that HA^x= A^xH for all ${A\in \mathcal{H}}$ and all ${x\in G}$. We say that a subgroup H of G is n-${\sigma}$-embedded in G if there exists a normal subgroup T of G such that HT is ${\sigma}$-permutable in G and ${H\cap T\leq H_{\sigma G}}$, where ${H_{\sigma G}}$ is the subgroup of H generated by all those subgroups of H which are ${\sigma}$-permutable in G.

In this paper, we study the properties of the n-${\sigma}$-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.