Imposing pseudocompact group topologies on Abelian groups

The least cardinal λ such that some (equivalently: every) compact group with weight α admits a dense, pseudocompact subgroup of cardinality λ is denoted by m(α). Clearly, m(α) ≤ 2. We show: Theorem 3.3. Among groups of cardinality γ, the group ⊕γQ serves as a “test space” for the availability of a pseudocompact group topology in this sense: If m(α) ≤ γ ≤ 2 then ⊕γQ admits a (necessarily connected) pseudocompact group topology of weight α ≥ ω (and also a pseudocompact group topology of weight log γ). Theorem 4.12. Let G be Abelian with |G| = γ. If either m(α) ≤ α and m(α) ≤ r0(G) ≤ γ ≤ 2, or α > ω and α ≤ r0(G) ≤ 2, then G admits a pseudocompact group topology of weight α. Theorem 4.15. Every connected , pseudocompact Abelian group G with wG = α ≥ ω satisfies r0(G) ≥ m(α). Theorem 5.2(b). If G is divisible Abelian with 2r0(G) ≤ γ, then G admits at most 2-many pseudocompact group topologies. Theorem 6.2. Let β = α or β = 2 with β ≥ α, and let β ≤ γ < κ ≤ 2 . Then both ⊕γQ and the free Abelian group on γ-many generators admit exactly 2-many pseudocompact group topologies of weight κ. Of these, some κ+-many form a chain and some 2-many form an anti-chain. 1991 Mathematics Subject Classification: Primary 54A05; Secondary 20K45, 22C05.