is a condition consistent with ZFC that follows from (but is much weaker than) the *Generalized Continuum Hypothesis* (GCH). Under SCH every infinite cardinal $\alpha$ satisfies $$m(\alpha)=(\log(\alpha))^\omega.$$ It is well known that every compact group has cardinality $2^\kappa$ for some cardinal $\kappa$. The question on which cardinals can appear as the cardinal of a pseudocompact group is not so readily answered. We will say that a cardinal $\kappa$ is *admissible* provided there is a pseudocompact group of cardinal $\kappa$. The first obstructions to admissibility were found by van Douwen [@douw80], the main one being that the cardinality $|G|$ of a pseudocompact group cannot be a strong limit cardinal of countable cofinality; see [@dikrshak98 Chapter 3] for more information on admissible cardinals.

Most of our results concern constructing pseudocompact group topologies on a given Abelian group $G$. As indicated in the introduction, every pseudocompact group topology is totally bounded and a totally bounded group topology $\mathcal{T}$ on an Abelian group $G$ is always induced by a unique group of characters $H\subset Hom(G,{{\mathbb T\,}})$, [@comfross64; @comfross66]. To stress this latter fact we will usually refer to $\mathcal{T}$ as $\mathcal{T}_{_{H}}$. Recall that the topology $\mathcal{T}_{_{H}}$ is Hausdorff if, and only if, the subgroup 