On pure subgroups of Cartesian products of integers
type: Item
instance of: scholarly article
title: On pure subgroups of Cartesian products of integers
zbMATH Open document ID: 0671.20052
DOI: 10.1007/BF03322445
author: Manfred Dugas
author: John Irwin
published in: Results in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: Denote by P the cartesian product of countably many copies of Z. The authors prove a number of results concerning pure subgroups of P such as pure embeddings of separable groups into P: 1. If H is a separable group with countable basic subgroup F, then it is purely embeddable into P in such a way that the generators of F are mapped upon the coordinate generators of P. 2. There exists a pure \(\aleph_ 1\)-separable (i.e. every countable subgroup is contained in a countable free summand) subgroup of P of cardinality \(\aleph_ 1\). 3. Every pure free subgroup of P is contained in a basic subgroup of P and there exist \(2^{2^{\aleph_ 0}}\) basic subgroups of P. 4. Under the additional axioms \(MA+\aleph_ 1<2^{\aleph_ 0}\), all \(\aleph_ 1\)-separable groups of cardinality \(\aleph_ 1\) can be embedded as pure subgroups into P. If, instead, the axiom \(\diamond_{\aleph_ 1}\) is assumed, then there is an \(\aleph_ 1\)-separable group of cardinality \(\aleph_ 1\) not embeddable in P. A consequence is that it cannot be decided in ZFC whether all \(\aleph_ 1\)-separable groups of cardinality \(\aleph_ 1\) can be embedded in P.
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 03E50
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 4097662
zbMATH Keywords: Baer-Specker group
zbMATH Keywords: undecidable
zbMATH Keywords: pure subgroups
zbMATH Keywords: pure embeddings
zbMATH Keywords: separable groups
zbMATH Keywords: pure free subgroup
zbMATH Keywords: basic subgroup
zbMATH Keywords: \(\aleph _ 1\)- separable groups
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322445
OpenAlex ID: W2079133756
description: scientific article; zbMATH DE number 4097662