Can a Compact Lie Group have a Non-Compact Lie Subgroup?

Question

Hopefully the title says it all in terms of the question I'm asking. I feel that the answer is no, but I'm not sure why; I don't really know many deep theorems about the structure of lie groups.

Answer 1

Depends on what you mean by "subgroup". If your definition of a Lie subgroup requires that the subgroup is closed or embedded (which is equivalent), then this is not possible. If not, then consider the example of one-parameter subgroup whose image is (after the obvious identifications) a straight line with an irrational slope inside the torus $S^1 \times S^1$.

Answer 2

Yes the real line is embedded as lie subgroup of the 2-dimensional torus.

Answer 3

I'm not going to worry about what the definition of "Lie subgroup" is, but a compact Lie group can certainly have a subgroup which is a non-compact Lie group. Consider the torus $$\Bbb T^2=\{(e^{it},e^{is}):t,s\in \Bbb R\}.$$ Suppose $\alpha\in\Bbb R$ is irrational and consider the subgroup $$\{(e^{it},e^{i\alpha t}):t\in\Bbb R\}.$$