Groups with many polycyclic-by-nilpotent subgroups
There is a common approach to study groups saturated with \(X\)-subgroups where \(X\) is some property pertaining to subgroups. In many articles groups with the minimal or maximal conditions for non-\(X\)-subgroups have been studied for some natural property \(X\) such as finite, Abelian, non-Abelian, subnormal, non-subnormal, nilpotent, hypercentral, locally nilpotent. In the article under review the authors consider locally graded groups satisfying the minimal condition on subgroups which are not polycyclic-by-finite-by-nilpotent. They also prove that a locally graded non-hypercentral group satisfying the minimal condition on non-hypercentral subgroups is periodic. Note, that a more detailed description of groups with all subgroups hypercentral and groups with the minimal condition on non-hypercentral subgroups one can find in the following articles, that have been accessible to at least one of the authors of the article under review. \textit{O. D. Artemovich}, Dopov. Akad. Nauk Ukr. 1997, No. 8, 7-9 (1997; Zbl 0933.20023), Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky 1998, No. 11, 7-9 (1998; Zbl 0933.20022), Ukr. Mat. Zh. 51, No. 10, 1425-1430 (1999; Zbl 0955.20016).
scientific article; zbMATH DE number 1462416