Classify the compact abelian Lie groups

Question

It's a classical theorem of Lie group theory that any compact connected abelian Lie group must be a torus. So it's natural to ask what if we delete the connectedness, i.e. the problem of classification of the compact abelian Lie groups.

Answer 1

The quotient of your Lie group $G$ by the connected component of the identity $G_0$ is a finite discrete abelian group $A$, because the Lie group is compact. It follows that we have an extension $$0\to G_0\to G\to A\to 0$$ Such a thing is classified (forgetting topologies) by an element of $H^2(A,G_0)$. This group is zero, because $A$ is finite and $G_0$ is divisible. It follows that $G\cong A\times G_0$. Since $G_0$ is a compact and connected, it is a torus.

This completely decribes the possible $G$s.