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

Document Type : Research Paper

Author

Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, Belarus

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 $i‎, ‎j\in \{1,\ldots‎,‎n\}$‎, ‎$i\neq j$‎, ‎is studied‎. ‎In particular‎, ‎we prove that if $G_i\in \frak F$ for all $i$‎, ‎then $G^\frak F\leq (G^\prime)^\frak N$‎, ‎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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