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Nearly \(s\)-embedded subgroups and the \(p\)-nilpotency of finite groups. (English) Zbl 1485.20055

Summary: A subgroup \(H\) of a finite group \(G\) is said to be nearly \(s\)-embedded in \(G\) if \(G\) has an \(s\)-permutable subgroup \(T\) and an \(s\)-semipermutable subgroup \(HssG\) contained in \(H\) such that \(H^{sG}=HT\) and \(H \cap T \le H_{ssG}\) where \(HsG\) is the intersection of all \(s\)-permutable subgroups of \(G\) containing \(H\). In this article, we investigate the influence of some nearly s-embedded subgroups of Sylow \(p\)-subgroups on the \(p\)-nilpotency of finite groups. A number of known results are improved and extended.

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D40 Products of subgroups of abstract finite groups
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