# Lie group action on manifold and the vector filed generated

Suppose $$G$$ is a Lie group and act smoothly on a compact oriented manifold $$M$$, $$X\in\mathfrak{g}$$ is an element in Lie algebra of $$G$$ such that the vector field $$\tilde X$$ generated by $$X$$ has isolated zeros, i.e.,

$$\tilde X(p)=\frac{d}{dt}\big\vert_{t=0}exp(tX)\cdot p$$

Then $$\tilde X$$ act on space of vector field on $$M$$ by Lie bracket: $$F\mapsto [\tilde X,F]$$. And this action induces a well-defined action on $$T_pM$$ for any point $$p$$ with $$\tilde X(p)=0$$. This action is denoted as

$$L(X,p):T_pM\to T_pM$$

$$~~~~~~~~~~~~~~~~~~~~~~v\mapsto [\tilde X,F_v](p)~~~~\text{where } F_v\text{ is a vector field with }F_v(p)=v$$

The question is to show that $$L(X,p)$$ is invertible.

I tend to show that if there is a non-zero vector $$\xi\in T_pM$$ s.t. $$L(X,p)(\xi)=0$$ then $$exp(tX)$$ fixes $$exp_p(sv)$$ for $$t,s$$ small. And this contradicts the fact that $$p$$ is isolated. But I don't know how to show this.