On the residual of a finite group with semi-subnormal 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 $B$. We introduce the new concept that unites subnormality and seminormality. A subgroup $A$ of a group $G$ is called semi-subnormal in $G$, if $A$ is subnormal in $G$ or seminormal in $G$. In this paper, the $\frak F$-residual of a group $G=AB$ with semi-subnormal subgroups $A$ and $B$ such that $A,B\in\frak F$, where $\frak F$ is a saturated formation and $\frak U\subseteq\frak F$, is studied. Here $\frak U$ is the class of all supersoluble groups and the $\mathfrak F$-residual of $G$ is the intersection of all those normal subgroups $N$ of $G$ for which $G/N\in\mathfrak F$.