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 # . thors 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 | ⩽ 2 c ; (c) r 0 ( G ) = | G | = | G | ω ; (d) | G | is a strong limit cardinal, and r 0 ( 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 r 0 ( G ) = | G | . Furthermore, the product of finitely many abelian G ∈ P ′ , each with the property T ∨ ( G ) = T # ( G ) , has the same property.