The Supersolvable Residual of a Finite Group Factorized by Pairwise Permutable Seminormal Subgroups

A subgroup A is seminormal in a finite group G if there exists a subgroup B such that G = AB and AX is a subgroup for each subgroup X from B. We study a group G = G1G2 . . .Gn with pairwise permutable supersolvable groups G1, . . . ,Gn such that Gi and Gj are seminormal in GiGj for any i, j ∈ {1, . . . , n}, i ≠ j. It is stated that GU = (G')N. Here N and U are the formations of all nilpotent and supersolvable groups, and HX and H' are the X-residual and the derived subgroup, respectively, of a group H. It is proved that a group G = G1G2 . . .Gn with pairwise permutable subgroups G1, . . .,Gn is supersolvable provided that all Sylow subgroups of Gi and Gj are seminormal in GiGj for any i, j ∈ {1, . . . , n}, i ≠ j.