Some open questions in the theory of generalized permutable subgroups.
All groups considered in the review are finite. A subgroup \(H\) of a group \(G\) is said to be s-permutable in \(G\) if it permutes with every Sylow subgroup of \(G\). \textit{A. N. Skiba} introduced [in J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)] the following generalizations of this concept: 1) A subgroup \(H\) of a group \(G\) is weakly s-permutable in \(G\) if there exists a subnormal subgroup \(T\) such that \(G=HT\) and \(T\cap H\subseteq H_{sG}\), where \(H_{sG}\) is the largest s-permutable subgroup of \(G\) contained in \(H\). 2) A subgroup \(H\) of a group \(G\) is said to be weakly s-supplemented in \(G\) if \(H\) has a supplement \(T\) in \(G\) such that \(H\cap T\leqslant H_{sG}\). In the above cited paper the author proved two theorems concerning the belonging of a group \(G\) to a saturated formation \(\mathcal F\) containing all supersoluble groups, under the assumption that certain subgroups of any non-cyclic Sylow subgroups of some normal subgroups of \(G\) are weakly s-permutable in \(G\). Next he stated as open questions whether the condition of weak s-permutability in those results could be replaced by the condition of weak s-supplementation in both results. In the paper under review the authors construct an example showing that both open questions have negative answers. Moreover, they obtain some positive results under some additional hypotheses regarding the weak s-supplementation of certain maximal (respectively, minimal) subgroups of Sylow subgroups.
scientific article; zbMATH DE number 5653990
scientific article