Given a Lie group $G$, $e$ is its identity element and $g$ is one element of $G$. I want do find a curve $\gamma(t)$ that satisfies these conditions:

1) passes $g$ and $e$, that is $\gamma(0)=e,\gamma(t\_g)=g$;

2) the tagent vector at $e$ is $v\in T\_eG$, that is $\gamma'(t)|\_{t=0}=\xi$.

Does this curve exist in a left invariant vector field? If yes, how to find it? If no, why? can we find it in other type of vector field?

My try: let $\gamma(t)=\exp(\xi t)$, so $\gamma(0)=e$, then solve $\gamma(t\_g)=\exp(\xi t\_g)=g$, get $t\_g$, then we have the vector at $g$: $\xi\_g=\gamma'(t)|\_{t=t\_g}$. I dont think my solution is right, but I dont know where is the problem, and how to get a better solution.

Thank you.