On σ-semipermutable Subgroups of Finite Groups

Let σ = {σ|i ∈ I } be some partition of the set of all primes P, G a finite group andσ(G) = {σ|σ∩π(G) = ?}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member = 1 of H is a Hall σ-subgroup of G for some σ∈σ and H contains exactly one Hallσ-subgroup of G for every σ∈σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if H H~x= H~xH for all x ∈ G and all H∈ H such that(|H|, |H|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.