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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.

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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