Forcing hereditarily separable compact-like group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2 c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2 c ) which admit: (i) ditarily separable group topology, p topology making G into an S-space, ditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), p topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary). y-product, we completely describe the algebraic structure of the Abelian groups of size at most 2 c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired). o put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.