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On solubility and supersolubility of some classes of finite groups. (English) Zbl 1189.20023
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 A. N. Skiba [J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)] and Y. Li, Y. Wang and H. Wei [Arch. Math. 81, No. 3, 245-252 (2003; Zbl 1053.20017)].

MSC:
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D25 Special subgroups (Frattini, Fitting, etc.)
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