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

Let G   be compact abelian group such that w(C(G))=w(Cω(G))w(C(G))=w(C(G))ω. We prove that if|C(G)|⩾m(G/C(G)),|C(G)|⩾m(G/C(G)), then G   contains a dense pseudocompact subgroup without non-trivial convergent sequences, where C(G)C(G) is the component of the identity of G   and m(G)m(G) is the smallest cardinality of a dense pseudocompact subgroup of G. As a consequence we obtain the following:(1)Every compact connected abelian group of weight κ   with κ=κωκ=κω has a dense pseudocompact subgroup without non-trivial convergent sequences.(2)[GCH] Let G be a compact abelian group whose connected component has weight κ   with κ=κωκ=κω. The following assertions are then equivalent:(i)Every dense pseudocompact subgroup of G has a non-trivial convergent sequence.(ii)One of the following two conditions is satisfied:(a)For some n<ωn<ω, nG   is infinite and cf(w(nG))=ωcf(w(nG))=ω.(b)|C(G)|