The fundamental group of Lie group

Question

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.

Answer 1

Since @the beatles link doesn't seem to work, I will try to answer, using references to results on compact Lie groups -

We can assume that $G$ is connected (since the fundamental group only sees a given connected component).

It is known that a compact Lie group $G$ is reductive, and so its lie algebra $\mathfrak g$ is of the form $\mathfrak g = \mathfrak z \oplus \mathfrak s$ where $\mathfrak z$ is central and $\mathfrak s$ is semi-simple.

$\mathfrak z$ is the Lie algebra of $Z(G^o)$, and so since the given group $G$ has trivial center it is in fact semi-simple. Finally, it is a theorem of Weyl that a semi-simple compact Lie group has finite fundamental group.

For the above results on compact Lie groups see e.g. chapter IV in Knapp's book Lie groups - beyond an introduction.