Abstract :
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.
Keywords :
Finite group , residual , seminormal subgroups , product of subgroups , derived subgroup
