International Journal of Group Theory (Mar 2022)
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 $i, j\in \{1,\ldots,n\}$, $i\neq j$, 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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