Supersolubility of a finite group with normally embedded maximal subgroups in Sylow subgroups

Abstract

A subgroup A is called seminormal in a group G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. Studying a group of the form G = AB with seminormal supersoluble subgroups A and B, we prove that GU = (G )N. Moreover, if the indices of the subgroups A and B of G are coprime then GU = GN2 . Here N, U, and N2 are the formations of all nilpotent, supersoluble, and metanilpotent groups respectively, while HX is the X-residual of H. We also prove the supersolubility of G = AB when all Sylow subgroups of A and B are seminormal in G.