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We describe an algebraic proof of the well-known topological fact that$\pi_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in ourapproach as the center of a certain finite group defined by generators andrelations. The latter is a factor group of the braid group $B_n$, obtained byimposing one additional relation and turns out to be a nontrivial centralextension by $Z/2Z$ of the corresponding group of rotational symmetries of thehyperoctahedron in dimension $n$.

The Fundamental Group of $SO(n)$ Via Quotients of Braid Groups