On σ-Permutably Embedded Subgroups of Finite Groups

Abstract

Let σ = {σ_i: i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σ_i: σ_i ∩ π(G)≠Ø}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ_i-subgroup of G and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ(G). G is said to be σ-full if G possesses a complete Hall σ-set. A subgroup H of G is σ-permutable in G if G possesses a complete Hall σ-set H such that HA^x= A^xH for all A ∈ H and all x ∈ G. A subgroup H of G is σ-permutably embedded in G if H is σ-full and for every σ_i ∈ σ(H), every Hall σ_i-subgroup of H is also a Hall σ_i-subgroup of some σ-permutable subgroup of G.

By using the σ-permutably embedded subgroups, we establish some new criteria for a group G to be soluble and supersoluble, and also give the conditions under which a normal subgroup of G is hypercyclically embedded. Some known results are generalized.