Euler characteristic of 2-dimensional compact Lie Groups

Question

I'd like to know why the Euler characteristic of $G$, a compact Lie Group of dimension 2, is zero. I'm aware of the fact that this is true not only for dimension 2. The point is that I'm not familiar with the techniques involved in the proofs I found so far. This question appears in the context of a first course in Riemannian Geometry, so regarding Lie Groups theory, I'm only familiar with the very basics definitions. I suppose it's easy for someone to explain me why is this true in the particular case of dimension 2. Thanks in advance :)

Answer 1

A (positive-dimensional) Lie group admits a non-vanishing vector field: Pick a non-zero element of the Lie algebra and use left translation to define an invariant vector field. (A similar argument shows the tangent bundle of $G$ is trivial.)

If $G$ is compact and $2$-dimensional, "existence of a non-vanishing vector field" implies "vanishing Euler characteristic" by the Poincaré-Hopf Index Theorem. (First apply the theorem to the identity component, then use the fact that any two components of $G$ are diffeomorphic.)

(One cannot generally conclude $G$ is a torus unless $G$ is also connected.)