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On \(\Phi\)-\(\tau\)-quasinormal subgroups of finite groups. (English) Zbl 1403.20028

Summary: Let \(\tau\) be a subgroup functor and \(H\) a \(p\)-subgroup of a finite group \(G\). Let \(\bar{G}=G/H_{G}\) and \(\bar{H}=H/H_{G}\). We say that \(H\) is \(\Phi\)-\(\tau\)-quasinormal in \(G\) if for some \(S\)-quasinormal subgroup \(\bar{T}\) of \(\bar{G}\) and some \(\tau\)-subgroup \(\bar{S}\) of \(\bar{G}\) contained in \(\bar{H}\), \(\bar{H}\bar{T}\) is \(S\)-quasinormal in \(\bar{G}\) and \(\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})\). In this paper, we study the structure of a group \(G\) under the condition that some primary subgroups of \(G\) are \(\Phi\)-\(\tau\)-quasinormal in \(G\). Some new characterizations about \(p\)-nilpotency and solubility of finite groups are obtained.

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
20D25 Special subgroups (Frattini, Fitting, etc.)
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