On groups with nilpotent by Černikov proper subgroups
type: Item
instance of: scholarly article
title: On groups with nilpotent by Černikov proper subgroups
zbMATH Open document ID: 0897.20021
author: Franco Napolitani
author: Elisabetta Pegoraro
published in: Archiv der Mathematik
publication date: 1998-08-17T00:00:00Z
review text: A group is called locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image. The authors study locally graded groups in which every proper subgroup is nilpotent-by-Chernikov. The main results are as follows. Theorem A: Let \(G\) be a locally graded group in which every proper subgroup is nilpotent-by-Chernikov. Then either \(G\) is nilpotent-by-Chernikov or else \(G\) is a perfect, countable, locally finite \(p\)-group with all proper subgroups nilpotent. Theorem B: Let \(G\) be a locally graded group. (i) If all proper subgroups are nilpotent-by-finite, then \(G\) is either nilpotent-by-finite or periodic. (ii) If all proper subgroups are abelian-by-finite, then \(G\) is either abelian-by-finite or periodic. Finally, the authors answer a question of Otal and Peña by establishing Theorem C: A locally graded group in which every proper subgroup is abelian-by-Chernikov is itself abelian-by-Chernikov.
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F22
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1145399
zbMATH Keywords: locally graded groups
zbMATH Keywords: nilpotent-by-Chernikov subgroups
zbMATH Keywords: finitely generated subgroups
zbMATH Keywords: locally finite \(p\)-groups
zbMATH Keywords: Abelian-by-finite groups
zbMATH Keywords: periodic groups
reviewed by: Derek J. S. Robinson
MaRDI profile type: Publication
full work available at URL: s000130050097
OpenAlex ID: W2025461722
DOI: 10.1007/S000130050097
description: scientific article; zbMATH DE number 1145399