On n- $${\sigma}$$ σ -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 HAx= AxH 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.