Hyper-Abelian groups with finite co-central rank.
type: Item
instance of: scholarly article
title: Hyper-Abelian groups with finite co-central rank.
zbMATH Open document ID: 1063.20037
author: Achim Tresch
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: As is well known, a group \(G\) has finite (Prüfer) rank if there is an integer \(r\geq 0\) such that every finitely generated subgroup of \(G\) can be generated by \(r\) elements. Here the author studies a more general rank restriction: a group \(G\) has finite co-central rank if there is an integer \(r\geq 0\) such that for every finitely generated subgroup \(H\) the quotient group \(H/Z(H)\) can be generated by \(r\) elements. In a previous work the author had shown that a locally soluble-by-finite group with finite co-central rank is hyper-(Abelian-by-finite). The converse of this result is proved here, so that the following holds. Theorem A. Let \(G\) be a group with finite co-central rank. Then \(G\) is locally soluble-by-finite if and only if it is hyper-(Abelian-by-finite). In addition it is shown that, not unexpectedly, the chief factors of the groups in Theorem A are finite, while their maximal subgroups have finite index.
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 2105406
zbMATH Keywords: hyperabelian groups
zbMATH Keywords: finite rank groups
zbMATH Keywords: groups of finite Prüfer rank
zbMATH Keywords: groups of finite co-central rank
zbMATH Keywords: locally soluble groups
zbMATH Keywords: finitely generated subgroups
reviewed by: Derek J. S. Robinson
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.03.001
OpenAlex ID: W2035871517
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DOI: 10.1016/J.JALGEBRA.2003.03.001
description: scientific article; zbMATH DE number 2105406