existence of infinite abelian subgroup in infinite locally compact groups

Question

1) Let $G$ be an infinite locally compact group.

Does there exist an infinite abelian locally compact subgroup of $G$?

Rem: I know that there exists an infinite abelian subgroup in every infinite compact group.

2) Does there exist structure theorems for locally compact groups which describe these groups with compact groups and abelian locally compact groups?

Answer 1

1. No. Consider the Tarski Monster as a discrete (hence locally compact) group. As all proper subgroups are finite, there are no infinite abelian subgroups.

2. If you're okay with ignoring discrete errors (i.e. looking at an open subgroup), then there is a sense in which all locally compact groups are "almost" Lie groups. For a precise statement, see the Gleason-Yamabe theorem which can be found on page 22 of http://terrytao.files.wordpress.com/2012/03/hilbert-book.pdf