A note on groups with a finite number of pairwise permutable seminormal subgroups
Publish place: International Journal of Group Theory، Vol: 11، Issue: 1
Publish Year: 1401
نوع سند: مقاله ژورنالی
زبان: English
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شناسه ملی سند علمی:
JR_THEGR-11-1_001
تاریخ نمایه سازی: 19 آبان 1400
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_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.
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Authors
Alexander Trofimuk
Department of Mathematics and Programming Technologies, Francisk Skorina Gomel State University, Gomel, Belarus