On pure subgroups of Cartesian products of integers
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.
scientific article; zbMATH DE number 4097662
scientific article