Mao, Yuemei; Ma, Xiaojian; Tang, Xingzheng; Huang, Jianhong On \(\Phi\)-\(\tau\)-quasinormal subgroups of finite groups. (English) Zbl 1403.20028 Bull. Iran. Math. Soc. 43, No. 7, 2169-2182 (2017). 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.) Keywords:\(S\)-quasinormal subgroups; \(p\)-nilpotent subgroups; subgroup functor; soluble group PDFBibTeX XMLCite \textit{Y. Mao} et al., Bull. Iran. Math. Soc. 43, No. 7, 2169--2182 (2017; Zbl 1403.20028) Full Text: Link