Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank).
It has been proved by \textit{F. Napolitani} and \textit{E. Pegoraro} [Arch. Math. 69, No. 2, 89-94 (1997; Zbl 0897.20021)] and \textit{A. O. Asar} [J. Lond. Math. Soc., II. Ser. 61, No. 2, 412-422 (2000; Zbl 0961.20031)] that if \(G\) is a locally graded group whose proper subgroups are nilpotent-by-Chernikov, then \(G\) itself is nilpotent-by-Chernikov. Recall that a group \(G\) is said to be `locally graded' if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index; thus locally graded groups form a wide class of generalized soluble groups, containing in particular all locally (soluble-by-finite) groups. The main result of the paper under review shows that if \(G\) is any locally graded group whose proper subgroups are Baer-by-Chernikov, then \(G\) itself is a Baer-by-Chernikov group. Here a group is called a `Baer group' if all its cyclic subgroups are subnormal.
scientific article; zbMATH DE number 5987919