Concerning Connected, Pseudocompact Abelian Groups

It is known that if P is either the property w-bounded or countably compact, then for every cardinal a 2 w there is a P-group G such that H.G = a and no proper, dense subgroup of G is a P-group. What happens when P is the property pseudocompact? The first-listed author and Robertson have shown that every zero-dimensional Abelian P-group G with H.G > o has a proper, dense, P-group. Turning to the case of connected P-groups, the present authors show the following results: Let G be a connected, pseudocompact, Abelian group with WG = a > W. If any one of the following conditions holds, then G has a proper, dense (necessarily connected) pseudocompact subgroup: (a) wG< c; (b) IGI 2 a”; (c) a is a strong limit cardinal and cf(a) > w; (d) ltor GI > c; (e) G is not divisible.