Dimension of image of one parameter subgroup.

Question

If $G$ is a Lie group then $\eta : \mathbb{R}\to G$ is called one parameter subgroup if it is a continuous group homomorphism.

I need to show that the images of one-parameter subgroups in a Lie group $G$ are precisely the connected Lie subgroups of dimension less than or equal to $1$.

I am not getting any idea how to start. Any hints are appreciated.

Thank you.

Answer 1

This is not true, consider the 2-dimensional torus $T^2$, there exists morphisms $\mathbb{R}\rightarrow T^2$ whose image are dense, such an image is not a subgroup since it is not closed.