On supersolubility of a group with seminormal subgroups

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