On generalised subnormal subgroups of finite groups. (Q5891238)

Let \(G\) be a finite group, \(\mathfrak F\) a formation of finite groups. A subgroup \(M\) of \(G\) is said to be \(\mathfrak F\)-normal (K-\(\mathfrak F\)-subnormal) in \(G\) if \(G/\text{Core}_G(M)\) belongs to \(\mathfrak F\) (there is a series \(U_0=M\leq U_1\leq\cdots\leq U_n=G\) with \(U_i\) normal or \(\mathfrak F\)-normal in \(U_{i+1}\)). These and related notions were studied, for instance, by \textit{A. Ballester-Bolinches} and \textit{M. D. Pérez-Ramos}, [Glasg. Math. J. 36, No. 2, 241-247 (1994; Zbl 0820.20023)], and \textit{A. Ballester-Bolinches, A. D. Feldman, M. C. Pedraza-Aguilera} and \textit{M. F. Ragland}, [J. Algebra 333, No. 1, 128-138 (2011; Zbl 1271.20014)].NEWLINENEWLINE The present paper investigates the question when K-\(\mathfrak F\)-subnormality implies \(\mathfrak F\)-normality, and gives answers under the following hypotheses: \(G\) is a PST-group, that is, Sylow permutability is a transitive relation among subgroups of \(G\), and \(\mathfrak F\) is closed under taking subnormal nonabelian subgroups of order \(pq\) of soluble PST-groups. Moreover, a characterization of soluble PST\(_{\mathfrak F}\)-groups is provided.