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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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