On σ-Embedded and σ-n-Embedded Subgroups of Finite Groups

Abstract

Let G be a finite group, and let σ = {σ_i | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σ_i | σ_i ∩ π(G) ≠ ∅}. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σ_i-subgroup of G and ℋ has exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ (G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HA^x = A^xH for all A ∈ ℋ and x ∈ G. A subgroup H of G is said to be σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H^G and H ∩ T ≤ H_σG, where H_σG is the subgroup of H generated by all those subgroups of H that are σ-permutable in G. A subgroup H of G is said to be σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H^σG and H ∩ T ≤ H_σG, where H^σG is the intersection of all σ-permutable subgroups of G containing H. We study the structure of finite groups under the condition that some given subgroups of G are σ-embedded and σ-n-embedded. In particular, we give the conditions for a normal subgroup of G to be hypercyclically embedded.