On Finite Groups Factorizable by Weakly Subnormal Subgroups

Abstract

Let $G$ be a group. A subgroup $H$ is weakly subnormal in $G$ if $H=\langle A,B\rangle$ for some subnormal subgroup $A$ and seminormal subgroup $B$ in $G$. Note that $B$ is seminormal in $G$ if there exists a subgroup $Y$ such that $G=BY$ and $AX$ is a subgroup for every subgroup $X$ in $Y$. We give some new properties of weakly subnormal subgroups and new information about the structure of the group $G=AB$ with weakly subnormal subgroups $A$ and $B$. In particular, we prove that if $A,B\in\mathfrak{F}$, then $G^{\mathfrak{F}}\leq(G^{\prime})^{\mathfrak{N}}$, where $\mathfrak{F}$ is a saturated formation such that $\mathfrak{U}\subseteq\mathfrak{F}$. Here $\mathfrak{N}$ and $\mathfrak{U}$ are the formations of all nilpotent and supersoluble groups correspondingly, and $G^{\mathfrak{F}}$ is the ${\mathfrak{F}}$-residual of $G$. Moreover, we study the groups $G=AB$ whose Sylow (maximal) subgroups from $A$ and $B$ are weakly subnormal in $G$.