On σ-semipermutable Subgroups of Finite Groups

Let σ = {σi|i ∈ I } be some partition of the set of all primes P, G a finite group andσ（G） = {σi |σi ∩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 σi-subgroup of G for some σi ∈σ and H contains exactly one Hallσi-subgroup of G for every σi ∈σ（G）. A subgroup H of G is said to be: σ-semipermutable in G with respect to H if H Hix= HixH for all x ∈ G and all Hi ∈ H such that（|H|, |Hi|） = 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.