Groups with all proper subgroups nilpotent-by-finite rank
As a continuation of their previous research the authors consider the groups in which every proper subgroup is nilpotent-by-finite rank. This article is rich of important results and also includes seven unsolved problems. For example, the authors prove that if a locally soluble-by-finite group \(G\) is a non-perfect \(p\)-group, has no infinite simple images, and every proper subgroup of \(G\) is nilpotent-by-finite rank, then \(G\) is itself nilpotent-by-finite rank. Among seven other theorems they also obtain the following interesting result. Let \(G\) be a locally soluble-by-finite group and suppose that every proper normal subgroup of \(G\) has finite rank. If \(G\) has no simple images, then \(G\) is radicable nilpotent and of finite rank.
nilpotent-by-finite rank proper subgroups
scientific article; zbMATH DE number 1538407