Let T is a hausdorff group topology and (G,T) is locally compact abelian group.If (G,T) has no open compact subgroups then can we say G has an infinite discrete cyclic subgroup?
This is a comment. I'm putting it as an answer just to draw attention to the question. (I hope it's ok.) It might be very easy. I'm very curious to know the answer. I'd state the question as follows:
Are there locally compact groups which have no compact open subgroups and no discrete infinite cyclic subgroups?
EDIT OF AUG. 1, 2010
By page 110 of Weil's book , the answer is No in the abelian case.
A particular case of the question in the nonabelian case is this. Let G be a non-compact connected Lie group. Does G admit a discrete infinite cyclic subgroup? Again, I suspect that this subquestion is very easy (at least for experts).
 Weil, André, L'intégration dans les groupes topologiques et ses applications, Actual. Sci. Ind., no. 869. Hermann et Cie., Paris, 1940. 158 pp.