A Lie group $H$ is called a Lie subgroup of a Lie Group $G$ if there is a map $i:H\to G$ which is (a) an injective immersion and (b) a group homomorphism.
My questios are: What happens if we replace (a) "injective immersion" by (a') "injective and differentiable"?
What happens if we go further and replace (a) "injective immersion" by (a'') "injective"?
Can anybody give examples where (a'') and (b) hold but not (a') and (b)? Or (a') and (b) but not (a) and (b)?
For the first question, note that, because $i$ is a group homomorphism, $i$ fails to be an immersion if and only if the derivative of $i$ degenerates at every point of $H$. If this happens, $i$ cannot be injective, by the constant rank theorem, for instance (note that, in fact, the rank is constant). Therefore, {injective + smooth + group hom} seem to imply {immersion}.
For the second question: A continuous homomorphism of Lie groups is automatically smooth (see the section titled continuous homomorphisms in Warner's Foundations of Diffferential Geometry and Lie Groups). Therefore, given a map $H \rightarrow G$ between Lie groups, {injective + continuous + group hom} seem to imply {$H \rightarrow G$ is a Lie subgroup}, which is pretty remarkable.