Residually nilpotent groups whose closed subgroups are subnormal.
A result of \textit{H. Smith} [J. Algebra 244, No. 2, 845-850 (2001; Zbl 0988.20018)] states that a periodic residually nilpotent group \(G\) in which all subgroups are subnormal is nilpotent. In this article the authors extend this result further. They take a particular residual nilpotency system for \(G\). By this is meant a family \(\mathcal N\) of normal subgroups of \(G\) with the properties: (1) \(G/N\) is nilpotent, for all \(N\in\mathcal N\); (2) for all \(N_1,N_2\in\mathcal N\) there exists \(N\in\mathcal N\) such that \(N\leq N_1\cap N_2\) and (3) \(\bigcap_{N\in\mathcal N}N=1\). A subgroup \(H\) of \(G\) is then said to be a closed subgroup of \(G\) if \(H=\bigcap_{N\in\mathcal N}HN\). The authors prove that if \(G\) is a periodic group admitting a residual nilpotency system with the property that all closed subgroups are subnormal then \(G\) is nilpotent. Interestingly, despite the natural topological idea involved, the proofs are for the most part group theoretic.
scientific article; zbMATH DE number 5949287