4 DIKRAN DIKRANJAN AND DMITRI SHAKHMATOV (ii) torsion-free Abelian groups, or even Abelian groups G with \G\ = r(G) (Theorem 1.4), (iii) torsion Abelian groups (Theorem 1.5), and (iv) divisible Abelian groups (Theorem 1.6). We also completely describe Abelian groups which admit a connected pseudo- compact group topology (Theorem 1.7). 1. PRINCIPAL RESULTS All necessary definitions and facts on varieties of abstract groups and properties of topological groups are collected in Section 2. The only essentially new result in this section is Lemma 2.17 which is used later in Section 3. Definition 2.6(H) introduces a purely set-theoretic condition P S ( T , a) between infinite cardinals r and cr, which was considered for the first time in 1978 by Cater, Erdos and Galvin [7]. This condition simply states that 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 innocently-sounding character, the condition Ps(r,cr) is not always easy to verify, and many questions related to it remain unsolved. (Some basic properties of PS(T,O~) are collected in Lemmas 2.7- 2.9). The importance of the condition PS(T,J) for our study is based on Fact 2.12 and Theorem 3.3(i), essentially due to Comfort and Robertson, which state that the set-theoretic condition Ps(r,cr) is equivalent to the existence of a pseudocompact group of cardinality r and weight a. We introduce Ps(r ) as an abbreviation for the statement U PS(T,a) holds for some infinite cardinal a". It is clear from the result of Comfort and Robertson that Ps(r ) is equivalent to the existence of a pseudocompact group of cardinality r. Therefore, Ps(|G|) is a necessary condition for the existence of a pseudocompact group topology on a group G. In Section 3 we discover a variety of other necessary conditions of this type, both

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