Connectivity, divisibility, and torsion

Abstract

Introduction.All of the groups discussed below are locally compact abelian groups, and additive notation is employed whenever convenient.A list at the end of the paper provides references to the pertinent definitions and notations.In §1 we define Z-groups by a constructive process.The basic idea is that of starting with the integers and the finite cyclic groups, and taking all groups which can be obtained by repeatedly including dual groups, including minimal divisible extensions, and including local direct products.§1 continues with some of the easier results on Z-groups, se\eral of which are listed below.§2 contains a discussion of certain important Z-groups, and some counterexamples.§3 discusses properties related to connectivity, divisibility, and torsion, and shows the significance which Z-groups have in the analysis of these properties.A characterization is obtained of the dual of a connected group.§4 gives a complete classification of homogeneous groups, and some results on endomorphism-simple groups.In §5 we conclude with some applications of the classification of homogeneous groups.These include a characterization of the dual of a torsion-free group, a classification of certain nonabelian automorphism groups, and results on the structure of division rings.Since Z-groups continue to appear throughout the various sections, we summarize the main results on Z-groups.Let Z7 be a locally compact abelian group which has compact identity component.Then 77 is a Z-group if any of the following conditions are satisfied : 77 discrete and finite, finitely generated, divisible, or free.77 torsion-free and compact or divisible.77 endomorphism-simple and a /j-group.77 the additive group of a locally compact field.77 homogeneous.Moreover, every connected 77 is a quotient group of a Z-group, every 77 which is torsion-free or endomorphism-simple is an open subgroup of a Z-group, and every H is a direct limit of groups which are inverse limits of Z-groups.The results on