On .-semipermutable Subgroups of Finite Groups

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