A B S T R A C T Looking for a natural generalization of compact spaces, in 1948 Hewitt intro- duced pseudocompact spaces as those Tychonoff spaces on which every real-valued continuous function is bounded. The algebraic structure of compact Abelian groups was completely described in the fifties and sixties by Kaplansky, Harrison and Hu- lanicki. In this paper we study systematically the algebraic structure of pseudocom- pact groups, or equivalently, the following problem: Which groups can be equipped with a pseudocompact topology turning them into topological groups? We solved this problem completely for the following classes of groups: free groups and free Abelian groups (or more generally, free groups in some variety of abstract groups), torsion- free Abelian groups (or even Abelian groups G with \G\ = r(G)), torsion Abelian groups, and divisible Abelian groups. Even though our main problem deals with the existence of some topologies on groups, it has a strong set-theoretic flavor. Indeed, the existence of an infinite pseudocompact group of cardinality r and weight a is equivalent to the following purely set-theoretic condition Ps(r, a) introduced by Cater, Erdos and Galvin for entirely different purposes: The set {0, l}a of all functions from (a set of cardinality) a to the two-point set {0,1} contains a subset of size r whose projection on every countable subproduct {0,1} A is a surjection. Despite its innocent look, the problem of which cardinals a and r enjoy such a relationship is far from being solved, and is closely related to the Singular Cardinal Hypothesis. A variety of necessary conditions, both of algebraic and of set-theoretic nature, for the existence of a pseudocompact group topology on a group is discovered. For ex- ample, pseudocompact torsion groups are locally finite. If an infinite Abelian group G admits a pseudocompact group topology of weight r, then either Ps(r(G), a) or Ps(rp(G), J) for some prime number p must hold, where r(G) and rp(G) are the free rank and the p-rank of G respectively. If an Abelian group G has a pseudocompact group topology, then \{ng : g € G}\ 22 for some n. This yields the inequality \G\ 22 for a divisible pseudocompact group. Turning to necessary and sufficient conditions, we show that a nontrivial Abelian group G admits a connected pseudocompact group topology of weight a if and only if \G\ 2a and Ps(r(G),o) hold. Moreover, a free group with r generators in a variety V of groups admits a pseudocompact group topology if and only if Ps(r , a) holds for some infinite r, and the variety V is generated by its finite groups. It should be noted that most of the classical varieties of groups have the last property, the only exception the authors are aware of being the Burnside varieties Bn for odd n 665. viii

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