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Finite groups with given $$s$$-embedded and $$n$$-embedded subgroups. (English) Zbl 1182.20026
From the summary: “Let $$G$$ be a finite group and $$H$$ a subgroup of $$G$$. Then $$H$$ is said to be $$s$$-permutable if $$H$$ permutes with all Sylow subgroups of $$G$$. Let $$H_{sG}$$ be the subgroup generated by all subgroups of $$H$$ which are $$s$$-permutable in $$G$$ and let $$H^{sG}$$ be the intersection of all $$s$$-permutable subgroups of $$G$$ which contain $$H$$. We say that: (1) $$H$$ is $$s$$-embedded in $$G$$ if $$G$$ has an $$s$$-permutable subgroup $$T$$ such that $$T\cap H\leq H_{sG}$$ and $$HT=H^{sG}$$; (2) $$H$$ is $$n$$-embedded in $$G$$ if $$G$$ has a normal subgroup $$T$$ such that $$T\cap H\leq H_{sG}$$ and $$HT=H^G$$.
Our main results here are the following: Theorem A. A group $$G$$ is supersoluble if and only if every maximal subgroup of every non-cyclic Sylow subgroup of the generalized Fitting subgroup $$F^*(G)$$ of $$G$$ is $$n$$-embedded.
Theorem B. A group $$G$$ is supersoluble if and only if for every non-cyclic Sylow subgroup $$P$$ of the generalized Fitting subgroup $$F^*(G)$$ of $$G$$, every cyclic subgroup $$H$$ of $$P$$ with prime order and with order 4 (if $$P$$ is a non-Abelian 2-group and $$H\not\subseteq Z_\infty(G)$$) is $$n$$-embedded.
Theorem F. A group $$G$$ is supersoluble if and only if every $$2$$-maximal subgroup $$E$$ of $$G$$ with non-primary index $$|G:E|$$, both has a cyclic supplement in $$E^{sG}$$ and is $$s$$-embedded in $$G$$.”
A number of published results appear then as corollaries of these theorems.

##### 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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