On Supersolubility of a Group with Seminormal 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 $G^{\mathfrak{U}}=\left(G^{\prime}\right)^{\mathfrak{N}}$. Moreover, if the indices of the subgroups A and B of G are coprime then $G^{\mathfrak{U}}=\left(G^{\mathfrak{N}}\right)^{2}$. Here $\mathfrak{N}$, $\mathfrak{U}$, and $\mathfrak{N}^2$ are the formations of all nilpotent, supersoluble, and metanilpotent groups respectively, while $H^{\mathfrak{X}}$ is the $\mathfrak{X}$-residual of H. We also prove the supersolubility of G = AB when all Sylow subgroups of A and B are seminormal in G.