Subgroup of a Lie group

Question

Is subgroup of a Lie group (in algebraic sense) automatically a manifold?

We all know an open subset of a manifold $M$ is also a manifold (but may not a submanifold sinse we havn't ask the inclusion map be an immersion or even embdedding). Since a Lie group $G$ is also a smooth manifold, so analogically, it is natural to ask whether its subgroup (in algebraic sense) also a manifold ?

Here let me first talk about my understanding :

Since we have Caratan's theorem, so if the subgroup $H$ of Lie group $G$ admits a manifold structure, then it satisfies the dinition of Lie group if and only if $H $ is a closed groupn since $H\times H \rightarrow H$ makes sense and it is smooth if we restricts the smooth map on $G_{|H}$.

Also I think it may lead some confusion by meaning "analogy" because open subset is a topological notion but not a group notion. So it seems to be strange to ask Lie subgroup meanwhile has a manifold structure which is a topological sense.

I'm not sure of my understanding. So any good example is welcome. Thanks for your suggestions !

Answer 1

Take the Lie group $(\mathbb{R},+)$, one may check that the subgroup $(\mathbb{Q},+)$ is not a manifold (with the subspace topology). Quotienting everything by $\mathbb{Z}$ gives a compact example.