Groups with the weak minimal condition for non-subnormal subgroups
type: Item
instance of: scholarly article
title: Groups with the weak minimal condition for non-subnormal subgroups
zbMATH Open document ID: 0939.20040
DOI: 10.1007/BF01783473
author: Howard Smith
author: L. A. Kurdachenko
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2000-01-04T00:00:00Z
review text: A descending chain \(H_1>H_2>\cdots\) of subgroups of a group \(G\) is called a descending \(\infty\)-chain if all the indices \(|H_n:H_{n+1}|\) are infinite. A group \(G\) is said to satisfying the weak minimal condition on non-subnormal subgroups if it has no infinite descending \(\infty\)-chains consisting of non-subnormal subgroups. Recall also that a group \(G\) is generalized radical if it has an ascending normal series whose factors are locally (nilpotent-by-finite). The authors prove that, if \(G\) is a generalized radical group satisfying the weak minimal condition on non-subnormal subgroups, then either \(G\) is a soluble-by-finite minimax group or all subgroups of \(G\) are subnormal.
Mathematics Subject Classification ID: 20F22
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 1383228
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: weak minimal condition on non-subnormal subgroups
zbMATH Keywords: descending \(\infty\)-chains
zbMATH Keywords: generalized radical groups
zbMATH Keywords: soluble-by-finite minimax groups
MaRDI profile type: Publication
cites work: Q4866213
cites work: Q4333325
cites work: Summands of Separable Abelian Groups: Dedicated to Paul Turán on his 60th Birthday
cites work: Groups with subnormality for all subgroups that are not finitely generated
cites work: Q5675882
cites work: fc-groups in which the orders of the elements of the periodic part are bounded
cites work: Groups satisfying weak minimum and maximum conditions for subnormal subgroups
cites work: Groups with weak minimality and maximality conditions for subgroups which are not normal
cites work: Q4883136
cites work: Q3744347
cites work: Q5517708
cites work: Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind. (Solubility of groups all subgroups of which are subnormal)
cites work: Q5654235
cites work: Q3941589
cites work: Q5656860
cites work: Some properties of groups inherited by normal subgroups of finite index
cites work: Theory of minimax groups
cites work: Q4200987
cites work: Q3876118
cites work: Properties of groups inherited by normal subgroups
full work available at URL: https://doi.org/10.1007/bf01783473
OpenAlex ID: W3085801718
description: scientific article; zbMATH DE number 1383228