subgroup of connected locally compact group

I need a reference or a short proof for the following property:

A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.

Solution 1:

I will assume, in addition, that $G$ is Hausdorff (otherwise the statement is clearly false). Then the statement is a direct corollary of the following

Theorem (Yamabe and Gleason). Let $G$ be a connected locally compact Hausdorff group. Then for every neighborhood $U$ of $1\in G$ there exists a compact normal subgroup $K<G$, so that $K\subset U$ and $L=G/K$ is isomorphic (as a topological group) to a (necessarily connected) Lie group.

See this post in Terry Tao's blog for the discussion of Y-G theorem. Note that, as Tao observes, there is no need in taking the subgroup $G'$ as we assume that $G$ is connected.

Now, given this, we note that, without loss of generality, $L$ is nontrivial (by taking $U$ small enough). A connected nontrivial Lie group always contains an infinite cyclic subgroup $Z$ (see Neal's comment). Now, take a generator $z\in Z$ and let $g\in G$ be its preimage under the epimorphism $G\to L$. Then the subgroup $<g>$ generated by $g$ is an infinite cyclic subgroup of $G$.