A criterion for \({\mathcal F}\)-subnormality (Q1117029)

The paper gives a criterion for \({\mathcal F}\)-subnormality, inspired by a subnormality criterion proposed by \textit{H. Wielandt} [Math. Z. 138, 199- 203 (1974; Zbl 0275.20041)]. This criterion is the following: For a subgroup H of a group G the following conditions are pairwise equivalent: (a) H is \({\mathcal F}\)-subnormal in G; (b) H is \({\mathcal F}\)-subnormal in \(<H,x>\), for every \(x\in G\); (c) H is \({\mathcal F}\)-subnormal in \(<H,H^ x>\), for every \(x\in G\); (d) from \(x\in G\) and \(x\in <H,H^ x>^{{\mathcal F}}\), it follows \(x\in H\), where all groups are assumed to be finite and soluble and \({\mathcal F}\) denotes a formation locally defined by the formation function f.