On torsion and mixed minimal Abelian groups.
type: Item
instance of: scholarly article
title: On torsion and mixed minimal Abelian groups.
zbMATH Open document ID: 1040.20043
author: S. ÓhÓgáin
author: B. Goldsmith
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT32-4.html
review text: As a way of investigating the concept of minimality in a general categorical setting, the authors call an Abelian group \(G\) minimal if it is isomorphic to every subgroup of finite index in \(G\). In 1998 Robinson and Timm explored this concept in non-Abelian group theory under the title of ``hc-groups''. In the present work, the authors obtain a characterization of torsion minimal groups in terms of their Ulm invariants, and characterizations of certain mixed torsion-free rank 1 minimal groups. In the torsion case, Goldsmith and hÓgáin show that a torsion group is minimal if and only if all of its primary components are minimal. Furthermore, if \(G\) is a reduced \(p\)-group, then \(G\) is minimal if and only if there exists an ordinal \(\lambda\leq\omega\) such that \(f_n(G)\) is infinite for all \(n<\lambda\) and, if \(\lambda<\omega\), \(f_n(G)=0\) for all \(n\geq\lambda\). The authors list several properties of minimal mixed groups and obtain several characterizations. One of them is that for certain torsion free rank 1 mixed groups \(G\), \(G\) is minimal if and only if \(t(G)\) and \(G/t(G)\) are minimal.
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 2012083
zbMATH Keywords: minimal Abelian groups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: Ulm invariants
zbMATH Keywords: mixed Abelian groups
reviewed by: Ross P. Abraham
MaRDI profile type: Publication
OpenAlex ID: W4229762797
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DOI: 10.1216/RMJM/1181070033
description: scientific article; zbMATH DE number 2012083