Classification of self-dual torsion-free LCA groups

In this paper we seek to describe the structure of self-dual torsion-free LCA groups. We first present a proof of the structure theorem of self-dual torsion-free metric LCA groups. Then we generalize the structure theorem to a larger class of selfdual torsion-free LCA groups. We also give a characterization of torsion-free divisible LCA groups. Consequently, a complete classification of self-dual divisible LCA groups is obtained; and any self-dual torsion-free LCA group can be regarded as an open subgroup of a well-understood torsion-free divisible LCA group. Introduction. After M. Rajagopalan and T. Soundararajan proved the structure theorem of self-dual torsion-free metric LCA groups in 1969, Corwin (1970) initiated a new and interesting approach to the problem of classifying the self-dual LCA groups in [2]. Though some sufficient and necessary conditions for an extension group G of a compact abelian group N by N̂ to be self-dual were given, the detailed structure of the group remains a mystery. In the last twenty years since the appearance of these two papers, no new progress appears to have been made. The problem of classifying self-dual LCA groups is still sitting in the dark, waiting for some light to be shed on it. We prove a structure theorem for self-dual torsion-free weak p-local LCA groups and present a complete classification of self-dual divisible LCA groups. The paper consists of six sections. Section 1 contains notations and basic definitions used in the paper. In Section 2, we give a brief discussion of direct product and prove a sufficient and necessary condition for a totally disconnected compact abelian group to be decomposed into a direct product of a family of its closed subgroups; this will be needed in Section 5 to characterize the local direct product ∑ i∈S(Ω i p : ∆ i p) of copies of the group Ωp of p-adic numbers. Section 3 gives a characterization of the group Ωp of p-adic numbers and shows that the direct product ∏ i∈S ∆ i p (or ∆ μ p in short, where μ = |S|) of copies of the group ∆p of p-adic integers is uniquely determined by the underlying index set and the prime number p. Section 4 presents a