# 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.

• $\mathbb{R}^\infty$ – Jimmy R Jun 13 '15 at 16:58

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.