(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 infinite compact subsets),
(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 infinite compact subsets if necessary).
As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired).
We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.