TITLE: Commutativity of the fundamental group of any Lie Group 
QUESTION [10 upvotes]: How do we formally prove that the fundamental group of any Lie group is always commutative?

REPLY [17 votes]: Geometric proof: A connected Lie group $G$ is homotopy equivalent to a maximal 
compact subgroup, so we may assume $G$ is compact. Being compact, $G$ admits a bi-invariant 
Riemannian metric with respect to which it is a symmetric space, the symmetry $s$ at the 
identity being just the inversion map. Now a homotopy class in $\pi_1(G,1)$ can be represented
by a closed geodesic $\gamma$ (of minimal length in its homotopy class, by a shortening process). Since the differential of $s$ at $1$ is minus identity, $s$ sends $\gamma$ to itself
parametrized backwards. It follows that the homomorphism induced by $s$ on the $\pi_1$-level is inversion. However, the inversion map in a group is a homomorphism if and only if the 
group is Abelian.