Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank).
type: Item
instance of: scholarly article
title: Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank).
zbMATH Open document ID: 1247.20044
author: Abdelhafid Badis
author: Nadir Trabelsi
published in: Central European Journal of Mathematics
publication date: 2011-12-12T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 5987919
zbMATH Keywords: locally graded groups
zbMATH Keywords: Baer groups
zbMATH Keywords: Baer-by-Chernikov groups
zbMATH Keywords: finitely generated subgroups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: subnormal subgroups
MaRDI profile type: Publication
full work available at URL: s11533-011-0077-0
OpenAlex ID: W2060142851
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DOI: 10.2478/S11533-011-0077-0
description: scientific article; zbMATH DE number 5987919