If a group is isomorphic to a Lie group will that group also be a Lie group?

Question

So I was working through some exercises on Lie groups and I was wondering if the group isomorphisms carry any of the differentiable structure with them.

Explicitly, if two groups, $G$ and $H$ are isomorphic and $G$ is a Lie group of dimension n can we conclude anything about the other group?

The group isomorphism, $\phi$, is a bijective function and so we know that the set theoretic structure of the spaces are the same, so we can put a topology on $H$ using the images of open sets in $G$ under the isomorphism (so the new topology will still be Hausdorff and second countable). This automatically makes $\phi$ into a homeomorphism and so we can then define charts using the charts of $G$ composed with $\phi$ and these will be $C^{\infty}$-compatible since we have just stuck a bijection and it's inverse into the middle when checking two charts are compatible. Then I believe that multiplication and inverses in $H$ are now smooth since when checking smoothness by composing with charts we have again just inserted a bijection and it's inverse into the composition, which will not alter the result and so won't change smoothness.

I think that what I have come up with is correct but I wanted to check that I haven't missed something.

Answer 1

Your construction looks like it will work to the extent that it gives you a new topology on $H$. However, notice that this doesn't have to have anything to do with the topology you would a priori expect for $H$.

For example, $\mathbb R$ and $\mathbb C$ under addition are both Lie groups but of different dimension. However they are isomorphic as groups, since both have dimension $2^{\aleph_0}$ as vector spaces over $\mathbb Q$.

Answer 2

If you do not have any structure on the group $H$, and have an isomorphism $G\to H$, then yes you can shift any information of $G$ on $H$ via this isomorphism. It is more or less given by the definition of the structure that you put.

Basically, you are doing nothing, and are in fact working on $G$, which has a structure, and just give a different "name" to the elements of the groups, via the identification $G\to H$. You can even do this with only a bijection to a set, and then create by this way the group structure on the other set.