Groups that are not Lie Groups What are some examples of groups that can not be given a smooth structure such that they become a Lie Group?
Edit: To be a bit more specific, I was hoping that somebody could give an example of a finite dimensional topological group that is a topological manifold but does not admit a smooth structure making it into a Lie Group.
 A: Any group is a Lie group if you give it the discrete topology. The better question is whether a topological group has a smooth structure that makes it a Lie group.
Local compactness is obviously necessary (because you want finite dimensions), so any non-locally compact group will be an example.
Generally, most locally compact groups are Lie groups. This question is essentially Hilbert's 5th Problem, which has been solved: https://en.wikipedia.org/wiki/Hilbert's_fifth_problem
A: You can take any uncountable Cartesian product of rationals $Q$ with discrete topology. This have discrete topology again (pre-images of $\emptyset \times \emptyset \times \cdots \times \{x\} \times \emptyset \times \emptyset \cdots, x\in Q$) are one-point elements in the product). Thus it has dimension 0 as a topological manifold. It is a group. It is a topological group (addition is bi-continuous and inversion is continuous, because of the discrete topology). Moreover, it is locally compact. But it is not second countable.
