On finite groups generated by strongly cosubnormal subgroups (Q1867309)

The following notion was introduced in the reviewer's paper [J. Algebra 234, No. 2, 609-619 (2000; Zbl 0969.20018)]: Two subgroups \(A\) and \(B\) of a finite group are called `strongly cosubnormal' if and only if every subgroup of \(A\) is cosubnormal with every subgroup of \(B\) in the sense of \textit{H. Wielandt} [Arch. Math. 35, 1-7 (1980; Zbl 0413.20020)]. The following characterization theorem has been proved: \(A\) and \(B\) are strongly cosubnormal if and only if the commutator group \([A,B]\) is contained in the hypercenter of \(\langle A,B\rangle\). In the present paper the authors obtain several related characterizations of strong cosubnormality. In particular they investigate the relation to the concept of \(\mathfrak N\)-connected subgroups introduced by \textit{A. Carocca} [Proc. Edinb. Math. Soc., II. Ser. 39, No. 1, 37-42 (1996; Zbl 0853.20010)]. Furthermore, it is shown that finite groups generated by strongly cosubnormal subgroups also behave well with respect to (not necessarily saturated) formations containing the class \(\mathfrak N\) of finite nilpotent groups.