On σ-semipermutable Subgroups of Finite Groups

Let sigma = {sigma(i)vertical bar i is an element of I} be some partition of the set of all primes P, G a finite group and sigma(G) = sigma(i)vertical bar i boolean AND pi(G) not equal empty set. A set H of subgroups of G is said to be a complete Hall sigma-set of G if every member not equal 1 of H is a Hall sigma(i)-subgroup of G for some sigma(i) is an element of sigma and H contains exactly one Hall of-subgroup of G for every sigma(i )is an element of sigma(G). A subgroup H of G is said to be: a-semipermutable in G with respect to H if HHix = H-i(x) H for all x is an element of G and all H-i is an element of H such that (vertical bar H vertical bar, vertical bar H-i vertical bar) = 1; sigma-semipermutable in G if H is sigma-semipermutable in G with respect to some complete Hall sigma-set of G. We study the structure of G being based on the assumption that some subgroups of G are sigma-semipermutable in G.