On the supremum of the pseudocompact group topologies

P is the class of pseudocompact Hausdorff topological groups, and P′ is the class of groups which admit a topology T such that (G,T ) ∈ P. It is known that every G= (G,T ) ∈ P is totally bounded, so for G ∈ P′ the supremum T ∨(G) of all pseudocompact group topologies on G and the supremum T #(G) of all totally bounded group topologies on G satisfy T ∨ ⊆ T #. The authors conjecture for abelian G ∈ P′ that T ∨ = T #. That equality is established here for abelian G ∈ P′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G| 2c; (c) r0(G) = |G| = |G|ω; (d) |G| is a strong limit cardinal, and r0(G)= |G|; (e) some topology T with (G,T ) ∈ P satisfies w(G,T ) c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r0(G)= |G|. Furthermore, the product of finitely many abelian G ∈ P′, each with the property T ∨(G)= T #(G), has the same property. © 2007 Elsevier B.V. All rights reserved. MSC: primary 22H11; secondary 54A25