Groups in which all subgroups are subnormal-by-finite (Q2818389)

The investigation of groups in which all subgroups are subnormal has resulted in some truly outstanding work in group theory, one of the highlights being the theorem of \textit{W. Möhres} [Arch. Math. 54, No. 3, 232--235 (1990; Zbl 0663.20027)] that such groups are soluble. Furthermore, \textit{C. Casolo} [J. Group Theory 5, No. 3, 292--300 (2002; Zbl 1002.20016)] has shown that locally finite groups, all of whose subgroups are subnormal, are in fact nilpotent-by-Chernikov. \textit{H. Heineken} [Note Mat. 16, No. 1, 131--143 (1996; Zbl 0918.20019)] investigated groups all of whose subgroups are subnormal-by-finite, where a subgroup \(H\) of a group \(G\) is subnormal-by-finite if there is a subnormal subgroup \(S\) of \(G\), lying in \(H\) and of finite index in \(H\). Groups in which every subgroup is normal-by-finite have been studied by \textit{J. T. Buckley} et al. [J. Aust. Math. Soc., Ser. A 59, No. 3, 384--398 (1995; Zbl 0853.20023)] where it is proved that locally finite such groups are abelian-by-finite. Heineken proved among other things that locally finite groups all of whose subgroups are subnormal-by-finite are locally nilpotent-by-finite. In the current paper, the author proves that if \(G\) is locally finite and every subgroup of \(G\) is subnormal-by-finite, then \(G\) is nilpotent-by-Chernikov and there is a natural number \(d\) such that every subgroup of \(G\) admits a subgroup of finite index which is subnormal of defect at most \(d\) in \(G\). The very elegant proofs make strong use of the work of Möhres. The paper ends with a brief discussion of some unsolved problems.