TITLE: Locally compact abelian groups
QUESTION [8 upvotes]: 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.

Now: 
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?

REPLY [8 votes]: Corollary 7.54 in Hoffman and Morris, The Structure of Compact Groups, seems to be what you want.

REPLY [6 votes]: I am not too familiar to Lie groups, but I think that your claim can follow from the following Theorem which is proven in Rudin:
If A is locally compact abelian group, then there exists a loc.cpct.abelian group $H$ such that $A$ is isomorphic to some $R^k \times H$, and $H$ contains an open compact subgroup $K$.
Reference: This is on page 95 in Deitmar Echterhoff "Principles of Harmonic Analysis", see also When does a LCA group not contain a (closed) infinite cyclic subgroup?.