On solubility and supersolubility of some classes of finite groups.
Let \(G\) be a finite group. A subgroup \(H\) is said to be S-permutable if \(PH=HP\) for every Sylow-subgroup \(P\). For a subgroup \(H\) let \(H_{sG}\) be the subgroup of \(H\) generated by all S-permutable subgroups of \(H\). A subgroup \(H\) is said to be S-embedded if for a normal subgroup \(T\) of \(G\) with \(T\cap H\leq H_{sG}\) we have \(HT\) S-permutable. The authors apply these notions for the description of supersoluble groups in the main result of the paper: \(G\) is supersolvable if and only if for every noncyclic subgroup \(P\) of the generalized Fitting subgroup of \(G\) either maximal subgroups of \(P\) are S-embedded, or, if \(P\) is a nonabelian 2-group and \(H\) is not contained in the hypercenter of \(G\), every cyclic subgroup of \(P\) of prime order or of order 4 is S-embedded. This is proved by means of three propositions of independent interest: these characterize solubility, and give sufficient conditions under which the group \(G\) belongs to a saturated formation containing all supersolvable groups using the above-mentioned notions. The strength of the main theorem and the auxiliary propositions is emphasized by many corollaries from which results of several recent research papers follow such as \textit{A. N. Skiba} [J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)] and \textit{Y. Li, Y. Wang} and \textit{H. Wei} [Arch. Math. 81, No. 3, 245-252 (2003; Zbl 1053.20017)].
scientific article; zbMATH DE number 5643408
scientific article