Injectivity of the exponential map of compact connected Lie groups

Question

I am trying to learn Lie Algebra with a little knowledge of Differential Geometry. I have read that the exponential map of compact connected Lie groups is always surjective. So I am wondering about the injectivity of the exponential map for compact connected Lie groups. I think it can not be injective and the multiplication of 1-torus with 1-sphere($T^1\times S^1$) is a counter example. Is this true? If so how can I prove it in a proper way?

Answer 1

If $G$ is compact, then $G$ has a subgroup isomorphic to the circle, therefore the exponential is not injective.