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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 $r>0$.

Any suggestions? Thanks in advance!

userr777
  • 846
  • Closed subgroups of $\mathbb{R}$ are of the form $\mathbb{R}$ or $r\mathbb{Z}$ for $r\in\mathbb{R}$ only. You don't need to compute the exponential map at all. See this: https://math.stackexchange.com/questions/90177/subgroup-of-mathbbr-either-dense-or-has-a-least-positive-element – freakish Oct 15 '18 at 09:14
  • And this for more details: https://arxiv.org/pdf/1312.7067.pdf – freakish Oct 15 '18 at 09:21

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