This is exercise from Lee: Introduction to smooth manifolds.
Suppose $f \colon G \to H$ is homomorphism of Lie groups (real, finite-dimensional).
Q: Is image $Im(f) \subseteq H$ a Lie subgroup of H?
That is, is there topology and smooth structure on $Im(f)$ such that inclusion $Im(f) \hookrightarrow H$ is immersion, and such that induced operation $Im(f) \times Im(f) \hookrightarrow H \times H \to Im(f) \subseteq H$ is smooth.
The author of these notes says (last paragraph on the first page) that the answer is no, provides counterexample (dense line on torus), but the proof is ommited. Is it correct?
I need positive proof of that, to understand the following characterisation: Lie group admits faithfull finite-dimensional representation if and only if it is (isomorphic to) Lie subgroup of $GL(n,\mathbb R)$.
One more thing. It is easy to see that $Ker(f)$ is Lie subgroup of $G$ using nontrivial Closed subgroup theorem. Is there more direct proof?
Any help is very appreciated.