Groups with all proper subgroups nilpotent-by-finite rank
type: Item
instance of: scholarly article
title: Groups with all proper subgroups nilpotent-by-finite rank
zbMATH Open document ID: 0965.20019
DOI: 10.1007/PL00000436
author: Martyn Dixon
author: Martin J. Evans
author: Howard Smith
published in: Archiv der Mathematik
publication date: 2001-07-05T00:00:00Z
review text: As a continuation of their previous research the authors consider the groups in which every proper subgroup is nilpotent-by-finite rank. This article is rich of important results and also includes seven unsolved problems. For example, the authors prove that if a locally soluble-by-finite group \(G\) is a non-perfect \(p\)-group, has no infinite simple images, and every proper subgroup of \(G\) is nilpotent-by-finite rank, then \(G\) is itself nilpotent-by-finite rank. Among seven other theorems they also obtain the following interesting result. Let \(G\) be a locally soluble-by-finite group and suppose that every proper normal subgroup of \(G\) has finite rank. If \(G\) has no simple images, then \(G\) is radicable nilpotent and of finite rank.
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E34
zbMATH DE Number: 1538407
zbMATH Keywords: nilpotent-by-finite rank proper subgroups
zbMATH Keywords: locally soluble-by-finite groups
zbMATH Keywords: \(p\)-groups
zbMATH Keywords: radicable nilpotent groups
reviewed by: Igor Subbotin
MaRDI profile type: Publication
full work available at URL: pl00000436
OpenAlex ID: W2040037920
description: scientific article; zbMATH DE number 1538407