On σ-semipermutable Subgroups of Finite Groups

Abstract

Let σ = {σi|i ∈ I} be some partition of the set of all primes P, G a finite group and σ(G) = {σi|σi ∩ π(G) 6= ∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member 6= 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exact one Hall σi-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 HHi x = Hi xH 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.