Lie group (additional condition)

[namlessom]

Does somebody know an example of a differentiable manifold which is a group but NOT a Lie group? So the additional condition: the group operations multiplication and inversion are analytic maps, is not satisfied.

[matt grime]

Yes, but they aren't particularly natural.

Let M be any differential manifold. Let G be any group such that card(M)=card(G). Let f be any bijection of the underlying sets. Then F makes M into a group by fiat, and in general, if we pick f in completely arbitrary fashion, it will not be a lie group.

More natural ones do not immediately spring to mind, sorry.