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 Abelian groups G (necessarily of size at most 2^c) which admit:

(i) a hereditarily separable group topology,

(ii) a group topology making G into an S-space,

(iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no non-trivial convergent sequences),

(iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without non-trivial convergent sequences, if necessary).

As a by-product, we completely describe the algebraic structure of Abelian groups of size at most 2^c which possess, at least consistently, a countably compact group topology (without non-trivial convergent sequences, if desired).

We also put to rest a 1980 problem of van Douwen about the cofinality of size of countably compact Abelian groups.

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