A Distinguished Subgroup of Compact Abelian Groups

Here “group” means additive abelian group. A compact group G contains δ–subgroups, that is, compact totally disconnected subgroups ∆ such that G/∆ is a torus. The canonical subgroup ∆(G) of G that is the sum of all δ–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of ∆(G) when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following: ∆(G) contains tor(G), is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and G/∆(G) is torsion-free and divisible; ∆(G) is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup ∆(G) appeared before in the literature as td(G) motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor, preserves proper exactness of short sequences of compact groups.