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

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_۱ G_۲ \cdots G_n with pairwise permutable subgroups G_۱‎,‎\ldots‎,‎G_n such that G_i and G_j are seminormal in~G_iG_j for any i‎, ‎j\in \{۱,\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.

کلیدواژه ها:

‎Finite group‎ ، ‎residual‎ ، ‎seminormal subgroups‎ ، ‎product of subgroups‎ ، ‎derived subgroup