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Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank). (English) Zbl 1247.20044

It has been proved by F. Napolitani and E. Pegoraro [Arch. Math. 69, No. 2, 89-94 (1997; Zbl 0897.20021)] and 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.

MSC:

20F19 Generalizations of solvable and nilpotent groups
20E07 Subgroup theorems; subgroup growth
20E25 Local properties of groups
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