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If $G$ is a compact Lie group whose Lie algebra $g$ has a trivial center, please show that the fundamental group of $G$ is finite.

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    Proposition 1.9, which states that the fundamental group of any topological manifold is countable, might be helpful: http://www.math.washington.edu/~lee/Books/Smooth/c01.pdf2011-11-29
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    Take a look at this: http://match.stanford.edu/lie/i22_1.html2011-11-29
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    Also the fact that the fundamental group of any topological group is Abelian may be useful.2013-02-24

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