Tangent space of lie subgroup

Question

If $H$ is a Lie subgroup of $G$ (and $H$ not necessarily a submanifold of $G$), would that still mean that $T_{e}H$ is a subspace of $T_{e}G$? My lecture notes define a Lie subgroup as just a subgroup that is a manifold, without any requirement that the smooth structure on $H$ is compatible with the smooth structure on $G$.

Answer 1

The standard definition of a Lie subgroup of a Lie group $G$ is a subset $H\subseteq G$ equipped with a Lie group structure such that the inclusion map $i:H\to G$ is an immersion and a group homomorphism. In particular, since $i$ is an immersion, it induces an injection $di_e:T_eH\to T_eG$ and so $T_eH$ can naturally be considered as a subspace of $T_eG$.

If you don't require the inclusion map to be an immersion, then all sorts of horrible things can happen. For instance, $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abstract groups, so you could take $G=\mathbb{R}$ with its usual manifold structure and $H=\mathbb{R}$ with a manifold structure pulled back along a group isomorphism $\mathbb{R}\to\mathbb{R}^2$. Then $T_eH$ has larger dimension that $T_eG$, so it certainly cannot be considered as a subspace in any way.