Direct sums and products of isomorphic Abelian groups
type: Item
instance of: scholarly article
title: Direct sums and products of isomorphic Abelian groups
zbMATH Open document ID: 0628.20046
DOI: 10.1216/RMJ-1987-17-3-573
author: John D. O'Neill
published in: Rocky Mountain Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The result of this note is the following: If G is a reduced abelian group, I, J are infinite sets and either \(| I|\) or \(| G|\) is non-measurable, then \(G^ I=\oplus_ JA\), for some subgroup A of G if and only if \(G=B\oplus C\), where \(B^ I\cong \oplus_ JT\), for some bounded subgroup T and \(C^ I\cong \oplus_ JC\cong C^ k\), for some positive integer k. An example of a reduced unbounded group G, such that \(G\cong G^ I\cong \oplus_ JG\) (I, J infinite sets) is obtained. If I is a set of measurable cardinality, then, for every group G, \(G^ I=L\oplus M\), where \(L\cong G\), \(M\cong G^ I\) and \(\oplus_ IG\) is a genuine subgroup of M.
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 4021458
zbMATH Keywords: reduced abelian group
zbMATH Keywords: non-measurable
MaRDI profile type: Publication
full work available at URL: rmj-1987-17-3-573
OpenAlex ID: W1978487522
description: scientific article; zbMATH DE number 4021458