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On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups. (English) Zbl 1205.20027
Summary: Let \(G\) be a finite group. A subgroup \(A\) of \(G\) is said to be \(S\)-quasinormal in \(G\) if \(AP=PA\) for all Sylow subgroups \(P\) of \(G\). The symbol \(H_{sG}\) denotes the subgroup generated by all those subgroups of \(H\) which are \(S\)-quasinormal in \(G\). A subgroup \(H\) is said to be \(S\)-supplemented in \(G\) if \(G\) has a subgroup \(T\) such that \(T\cap H\leqslant H_{sG}\) and \(HT=G\); [see A. N. Skiba, J. Algebra 315, No. 1, 192-209 (2007; Zbl 1130.20019)].
Theorem A. Let \(E\) be a normal subgroup of a finite group \(G\). Suppose that for every non-cyclic Sylow subgroup \(P\) of \(E\), either all maximal subgroups of \(P\) or all cyclic subgroups of \(P\) of prime order and order 4 are \(S\)-supplemented in \(G\). Then each \(G\)-chief factor below \(E\) is cyclic.
Theorem B. Let \(\mathcal F\) be any formation and \(G\) a finite group. If \(E\vartriangleleft G\) and \(F^*(E)\leqslant Z_{\mathcal F}(G)\), then \(E\leqslant Z_{\mathcal F}(G)\).
These theorems give positive answers to two questions of Shemetkov and strengthen results of various authors.

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
20D30 Series and lattices of subgroups
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