The supersolvable residual of a finite group factorized by pairwise permutable seminormal subgroups

A subgroup $A$ is seminormal in a finite group $G$ if there exists a subgroup $B$ such that $G=AB$ and $AX$ is a subgroup for each subgroup $X$ from $B$. We study a group $G=G_1G_2\ldots G_n$ with pairwise permutable supersolvable groups $G_1,\ldots,G_n$ such that $G_i$ and $G_j$ are seminormal in $G_iG_j$ for any $i,j\in\{1,\ldots,n\}$, $i\neq j$. It is stated that $G^\mathfrak U=(G^\prime)^\mathfrak N$. Here $\mathfrak N$ and $\mathfrak U$ are the formations of all nilpotent and supersolvable groups, and $H^\mathfrak X$ and $H^{\prime}$ are the $\mathfrak X$-residual and the derived subgroup, respectively, of a group $H$. It is proved that a group $G=G_1G_2\ldots G_n$ with pairwise permutable subgroups $G_1,\ldots,G_n$ is supersolvable provided that all Sylow subgroups of $G_i$ and $G_j$ are seminormal in $G_iG_j$ for any $i,j\in\{1,\ldots,n\}$, $i\neq j$.