Abstract
A subgroup $A$ of a group $G$ is called {it 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 $B$. The group $G = G_1 G_2 cdots G_n$ with pairwise permutable subgroups $G_1,ldots,G_n$ such that $G_i$ and $G_j$ are seminormal in~$G_iG_j$ for any $i, jin {1,ldots,n}$, $ineq j$, is studied. In particular, we prove that if $G_iin frak F$ for all $i$, then $G^frak Fleq (G^prime)^frak N$, where $frak F$ is a saturated formation and $frak U subseteq frak F$. Here $frak N$ and $frak U$~ are the formations of all nilpotent and supersoluble groups respectively, the $mathfrak F$-residual $G^frak F$ of $G$ is the intersection of all those normal subgroups $N$ of $G$ for which $G/N in mathfrak F$.
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