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On \(\mathcal F\)-supplemented subgroups of finite groups. (English) Zbl 1172.20019
All groups involved in this review are finite. Recall that a formation is a class of groups which is closed under taking epimorphic images and subdirect products. If \(\mathcal F\) is a formation and \(G\) is a group, a subgroup \(H\) of \(G\) is said to be \(\mathcal F\)-supplemented in \(G\) if there exists a subgroup \(T\) of \(G\) such that \(G=TH\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathcal F\)-hypercenter \(Z^{\mathcal F}_\infty(G/H_G)\) of \(G/H_G\), that is, the largest normal subgroup of \(G/H_G\) for which every chief factor of \(G/H_G\) below it is \(\mathcal F\)-central in \(G/H_G\). Of course normal or complemented subgroups (more generally, c-supplemented subgroups introduced by X. Guo, Y. Wang and the reviewer [in Glasg. Math. J. 42, No. 3, 383-389 (2000; Zbl 0968.20009)]) are examples of \(\mathcal F\)-supplemented subgroups for every non-empty formation \(\mathcal F\).
In this paper, the influence of \(\mathcal F\)-supplemented subgroups on the structure of a group is analyzed. For instance, a group \(G\) is soluble if and only if every subgroup of prime order (respectively, every Sylow subgroup) is supplemented in \(G\) with respect to the formation of all soluble groups (Theorems 4.1 and 4.2). The author also proves that if \(\mathcal F\) is a subgroup-closed saturated formation containing all supersoluble groups and \(G\) is a group with a subgroup \(E\) containing the \(\mathcal F\)-residual of \(G\), then \(G\) belongs to \(\mathcal F\) provided that every subgroup of prime order or order \(4\) of \(E\) is supplemented in \(G\) with respect to the formation of all supersoluble groups. The same conclusion follows if the subgroups of prime order or order \(4\) are replaced by the family of all maximal subgroups of the non-cyclic Sylow subgroups of \(E\) not having a supersoluble supplement in \(G\). A series of known results about normal, c-normal and c-supplemented subgroups are unified and generalized.
Remarks: It should be noted that the concept of c-supplemented subgroup was introduced by Guo, Wang and the reviewer [in loc. cit.] and not by Y. Wang [in J. Algebra 224, No. 2, 467-478 (2000; Zbl 0953.20010)], as the author claims in the paper.

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