a note on groups with a finite number of pairwise permutable seminormal subgroups

A subgroup a of a group g is called {it seminormal} in g‎, ‎if there exists a subgroup b such that g=ab and ax~is a subgroup of g for every‎ ‎subgroup x of b‎. ‎the group g=g1g2⋯gn with pairwise permutable subgroups g1‎,‎…‎,‎gn such that gi and gj are seminormal in~gigj for any i‎,‎j∈{1,…‎,‎n}‎, ‎i≠j‎, ‎is studied‎. ‎in particular‎, ‎we prove that if gi∈f for all i‎, ‎then gf≤(g′)n‎, ‎where f is a saturated formation and u⊆f‎. ‎here n and u‎~ ‎are the formations of all nilpotent and supersoluble groups respectively‎, ‎the f-residual gf of g is the intersection of all those normal‎ ‎subgroups n of g for which g/n∈f‎.