I am reading introductory lecture notes on Lie groups and Lie algebras. There it is stated as a fact without proof, that any compact semi-simple Lie group has finite center.
Here, semi-simple means, that the corresponding Lie algebra can be written as a direct sum of simple Lie algebras (having no non-trivial ideals).
Since this is not immediately obvious for me, I wonder if the proof is actually complicated or if I am just too ignorant to see it. If anyone can give an idea of where this fact comes from, I would be thankful.
If $G$ is a semi-simple Lie group, then its Lie algebra $\frak g$ is semi-simple, which implies that it has trivial center. Therefore, the Lie algebra of the center $Z(G)$ of $G$ is trivial, which means that $Z(G)$ is discrete. Since $G$ is compact, $Z(G)$ is finite.