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

Let G be a finite group, and let σ = {σi | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σi | σi ∩ 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 σi-subgroup of G and ℋ has exactly one Hall σi-subgroup of G for every σi ∈ 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 HAx = AxH 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 = HG and H ∩ T ≤ HσG, where HσG 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 = HσG and H ∩ T ≤ HσG, where HσG 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.