When $\alpha_G$ is a topological isomorphism we say that $G$ is Pontryagin reflexive. It will be necessary for the development of this paper to keep in mind that character groups of discrete groups are compact groups. Even if it is not relevant for our purposes we cannot resist here to add that character groups of compact groups are again discrete, and that the Pontryagin van-Kampen theorem proves that all locally compact Abelian groups (discrete and compact ones are thus comprised) are reflexive.

In the present paper Pontryagin duality will appear both as a tool for constructing pseudocompact group topologies and as an objective itself. To be precise, this paper is motivated by the following two questions

\[Qref\] Is every Pontryagin reflexive totally bounded Abelian group a compact group?

\[Qcomp\] Does every pseudocompact Abelian group admit a pseudocompact group topology with no infinite compact subsets?

In this paper we obtain a negative answer to Question \[Qref\] and a positive answer, valid under the Singular Cardinal Hypothesis (SCH), to Question \[Qcomp\]. The focus of the paper will be on Question \[Qcomp\] with the analysis of Question \[Qref\] and its relation with Question \[Qcomp\] deferred to Section 6.

It should be noted, in 