Concerning connected, pseudocompact Abelian groups

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Abstract

It is known that if P is either the property ω-bounded or countably compact, then for every cardinal α ⩾ ω there is a P-group G such that wG = α 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 wG >ω 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 = α >ω. If any one of the following conditions holds, then G has a proper, dense (necessarily connected) pseudocompact subgroup: (a) wG ⩽ c; (b) |G| ⩾ αω; (c) α is a strong limit cardinal and cf(α) >ω; (d) |torG| >c (e) G is not divisible.

MSC

54A05

MSC

20K45, 22C05

Keywords

pseudocompact group
connected group
dense subgroup

Cited by (0)

The second-listed author acknowledges partial support from the National Scince Foundatiion (USA) under grant NSF-DMS-8606113. He is pleased to thank also the Department of Mathematics at Wesleyan University for generous hospitality and support during the summers of 1986 and 1987.