A NOTE ON GROUPS WITH A FINITE NUMBER OF PAIRWISE PERMUTABLE SEMINORMAL SUBGROUPS

A subgroup A of a group G is called 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 = G(1)G(2)center dot center dot center dot G(n) with pairwise permutable subgroups G(1),...,G(n) such that G(i) and G(j) are seminormal in G(i)G(j) for any i; j is an element of {1,...,n}, i not equal j, is studied. In particular, we prove that if G(i) is an element of f for all i, then G(f) <= (G '')(n), where F is a saturated formation and u subset of f. Here n and u are the formations of all nilpotent and supersoluble groups respectively, the f-residual G(f) of G is the intersection of all those normal subgroups N of G for which G=N is an element of f.