A note on groups with a finite number of pairwise permutable seminormal subgroups

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 $G/N in mathfrak F$‎.