First, some preliminaries:
Define an "LCA group" to be a locally compact Hausdorff abelian topological group.
Define "smooth manifold" in a way that requires Hausdorffness, but not connectedness or paracompactness. Define a "Lie group" to be a smooth manifold with smooth group operations. Note that with these definitions, any discrete topological space is a manifold, and any discrete topological group is a Lie group.
I have been told that any LCA group A has a compact subgroup K such that A/K is a Lie group.
However, I have not been able to extract this result from the literature. For some attempts, see this post to the n-Category Cafe.
Can anyone find a proof of this result, or prove it?