Is the exponential map ever not injective?
Solution 1:
As a simple counterexample: The group $SO(2) \subset GL_2(\mathbb{R})$ is isomorphic to $\mathbb{S}^1$; its Lie algebra is the space of skew-symmetric matricies, and its exponential map is given by $$ e : \left( \begin{array}{cc} 0 & t \\ -t & 0 \end{array} \right) \mapsto \left( \begin{array}{cc} \cos t & \sin t \\ -\sin t & \cos t \end{array} \right) $$ which isn't injective.