Pseudocompact topological group refinements of maximal weight

It is known that a compact metrizable group admits no proper pseudocompact topological group refinement. The authors show, in contrast, that every (Hausdorff) pseudocompact Abelian group $G=(G,{\mathcal T})$ of uncountable weight $\alpha$, satisfying any of the following conditions, admits a pseudocompact group refinement of maximal weight (that is, of weight $2^{\vert G\vert}$):

(i)

$G$ is compact;

(ii)

$G$ is torsion-free with $\alpha\leq\vert G\vert=\vert G\vert^\omega$;

(iii)

[GCH] $G$ is torsion-free.

Remark. (i) answers a question posed by Comfort and Remus [Math. Zeit- schrift 215 (1994), 337-346].