Some topological properties of a Lie group can be deduced by studying a discrete group of homotopy classes of paths from the identity to elements of a finite subgroup of the given Lie group. In this way a "skeleton" of the universal cover is constructed in terms of generators and relations. We use this approach to describe an algebraic derivation of the well-known fact that the fundamental group of SO(n) is isomorphic to Z/2Z when n ≥ 3. The fundamental group of SO(n) appears in our treatment as a subgroup of the center of a finite factor of the braid group Bn, obtained by imposing one additional relation and turns out to be a nontrivial central extension by Z/2Z of the corresponding group of rotational symmetries of the hyperoctahedron in dimension n.
