# Connected Lie group 1-dimensional isomorphic to $\mathbb{R}$ or to $S^1$

Let $$G$$ a connected Lie group of dimension 1. Show that \begin{align} G \cong \mathbb{R} \, \, \, \text{or} \, \, \, G \cong S^1 \end{align}

I tried to read and understand the topic Connected, one-dimensional Lie groups , but I have some problem to compute the kernel of the exponential map to see that $$\ker(\exp) = \{ 0 \}$$ or $$\ker(\exp)= r\mathbb{Z}$$ for some $$r>0$$.