Compact groups containing dense pseudocompact subgroups without non-trivial convergent sequences (Q864468)

The authors enlarge the list of compact groups that admit a dense pseudocompact subgroup without non-trivial convergent sequences and point toward classification of them. Let \(G\) be a compact group. Denote by \(C(G)\) the component of the identity in \(G\), by \(m(G)\) the smallest cardinality of a dense pseudocompact subgroup of \(G\) and by \(w(G)\) the weight of \(G\). The authors show that for a compact abelian group \(G\) with \(w(C(G))=w(C(G))^\omega\) the condition \(| C(G)| \geq m(G/C(G))\) guarantees the existence of a dense pseudocompact subgroup without non-trivial convergent sequences. For an arbitrary abelian group \(G\), \(r(G)\) stands for its rank and \(r_0(G)\) denotes its torsion-free rank, i.e., the cardinality of a maximal independent subset of \(G\) consisting entirely of non-torsion elements. It follows that every connected abelian group of weight \(\kappa\) with \(\kappa=\kappa^\omega\) has a dense pseudocompact subgroup without non-trivial convergent sequences. The authors show under the assumption of GCH that whenever \(G\) is a compact abelian group whose connected component has weight \(\kappa\) with \(\kappa=\kappa^\omega\) the following assertions are equivalent: (i) Every dense pseudocompact subgroup of \(G\) has a non-trivial convergent sequence. (ii) One of the two conditions is satisfied: (a) For some \(n<\omega\), \(nG\) is infinite and \(cf(w(nG))=\omega\). (b) \(| C(G)| <m(\log(r_0(G/C(G))))\).