Groups in which every proper subgroup is Chernikov-by-nilpotent or nilpotent-by-Chernikov
type: Item
instance of: scholarly article
title: Groups in which every proper subgroup is Chernikov-by-nilpotent or nilpotent-by-Chernikov
zbMATH Open document ID: 0632.20018
DOI: 10.1007/BF01207469
author: J. M. Peña
author: Javier Otal
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: \textit{B. Bruno} and \textit{R. E. Phillips} [Rend. Semin. Mat. Univ. Padova 69, 153-168 (1983; Zbl 0522.20022)] have classified infinite groups in which every proper subgroup is finite-by-nilpotent of class \(c\) whereas \textit{B. Bruno} [Boll. Unione Mat. Ital., VI. Ser. B 3, 797-807 (1984; Zbl 0563.20035) and ibid. D 3, 179-188 (1984; Zbl 0578.20027)] has considered the ``dual'' situation studying the cases in which proper subgroups are Abelian-by-finite and nilpotent-by-finite. The results in the present paper extend the above problems by replacing finite group by Chernikov group and they have a rather different nature than those of Bruno and Phillips because the main theorems give subgroup characterizations of the properties under consideration. These theorems are the following 1) A locally graded group \(G\) is Chernikov-by-nilpotent of class \(c\) if and only if every proper subgroup of \(G\) is Chernikov-by-nilpotent of class \(c\). 2) A periodic locally graded group \(G\) is Abelian-by-Chernikov if and only if every proper subgroup of \(G\) is Abelian-by-Chernikov.
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 4027745
zbMATH Keywords: Chernikov groups
zbMATH Keywords: subgroup characterizations
zbMATH Keywords: locally graded groups
zbMATH Keywords: Chernikov-by-nilpotent groups
zbMATH Keywords: Abelian-by-Chernikov groups
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01207469
OpenAlex ID: W2063096648
description: scientific article; zbMATH DE number 4027745