a direction opposite to Question \[Qcomp\], that every pseudocompact group admits pseudocompact group topology with nontrivial convergent sequences, see [@galigarctomi09].

Our approach to Question \[Qcomp\] consists in combining techniques that can be traced back at least to [@tkac88] with the ideas of [@galigarc07]. Our construction actually produces pseudocompact Abelian groups with all countable subgroups $h$-embedded. This is stronger (see Section 2) that finding pseudocompact group topologies with no infinite compact subsets. With the aid of results from [@hernmaca03] this construction will yield a wide range of negative answers to Question \[Qref\]. As pointed to us by M. G. Tkachenko, Question \[Qref\] has been answered independently in [@ardaetal].

On notation and terminology {#on-notation-and-terminology .unnumbered}
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All groups considered in this paper will be Abelian. So, the specification *Abelian group* to be found at some points will respond only to a matter of emphasis. To further avoid the cumbersome use of the word “Abelian”, free Abelian groups will simply be termed as *free groups*.

The symbol ${{\mathbb P}}$ will denote the set of all prime numbers. *Faute de mieux*, we will use the unusual symbol $\mathbb{P}^{\uparrow}$ to denote the set of all prime powers, i.e., an integer $k\in \mathbb{P}^{\uparrow}$ if, and only 