Free subgroups with torsion quotients and profinite subgroups with torus quotients
All groups are abelian. The authors study the full free subgroups \(F\) of a torsion free abelian group of finite rank \(A\) i.e., the free subgroups of \(A\) with torsion quotient. Let \(\mathfrak{F}(A)\) denote the set of all full free subgroups of \(A\). It is obtained numerous properties of groups which belong to the class \(\mathfrak{F}(A)\). A protorus is a compact, connected, finite dimensional topological group. If \(G\) is a protorus then compact totally disconnected subgroups \(\Delta\) of \(G\) such that \(G/\Delta\) is a torus is called \(\delta\)-subgroup. \(\delta\)-subgroups are essential ingredients in the important Resolution Theorem a description of compact groups. The authors receive the characterization \(\delta\)-subgroups from the properties of full free subgroups with the help of Pontryagin's duality. The authors introduce a new topology on a torsion free abelian group \(A\) by the set \(\mathfrak{F}(A)\) (the free or \(\mathfrak{F}\)-topology of \(A\)). It is received some results about completion in this topology. The authors introduce by \(\delta\)-subgroups the new topology on a protorus (\(\mathcal{D}\)-topology) and they explore it.
scientific article; zbMATH DE number 7367416
scientific article