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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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