Vector field derived from Group action on smooth manifolds

Question

It is known that given a smooth action $G$ on $M$, $$G\times M\to M$$ one can associate to each element of $v\in T_eG$ a vector field on $X\in \mathfrak{X}(M)$. I want to see a less obvious and concrete example and know that is this fact just about Lie groups of positive dimension or it is meaningful for discrete groups action too?

Answer 1

Nothing goes wrong in the case of a discrete (i.e. $0$-dimensional) Lie group; it's just a rather trivial case: If $G$ is a discrete group, then the tangent space is a singleton $T_eG=\{0\}$, and $0\in T_eG$ corresponds to the zero vector field $0\in\mathfrak{X}M$.