On locally finite groups whose derived subgroup is locally nilpotent (Q6656188)

The aim of the paper under review is to extend some results on subnormal subgroups, which are known in the finite case, to the class of locally finite groups. In a seminal paper \textit{H. Wielandt} ([Math. Z. 45, 209--244 (1939; JFM 65.0061.02)]) proved the so-called join theorem that asserts that the subgroup generated by two subnormal subgroups of a finite group is subnormal.\N\NAlthough Wielandt's result is not true in arbitrary locally finite groups, the author is able to extend it to the case to homomorphic images of periodic linear groups. In addition, the author extends the results of [\textit{A. Ballester-Bolinches} et al., Math. Nachr. 239--240, 5--10 (2002; Zbl 1002.20014)] to locally finite groups, so it is possible to characterize the class of locally finite groups with a locally nilpotent derived subgroup as the largest subgroup-closed saturated formation \(\mathfrak{X}\) such that, for all \(\mathbf{SL}\)-closed saturated formations \(\mathcal{F}\), the \(\mathcal{F}\)-residual of an \(\mathfrak{X}\)-group generated by \(\mathcal{F}\)-subnormal subgroups is the subgroup generated by their \(\mathcal{F}\)-residuals.