  A few years ago Kramer and Laubenbacher introduced a discrete notion of
homotopy for simplicial complexes. In this paper, we compute the discrete
fundamental group of the order complex of the Boolean lattice. As it turns out,
it is equivalent to computing the discrete homotopy group of the 1-skeleton of
the permutahedron. To compute this group we introduce combinatorial techniques
that we believe will be helpful in computing discrete fundamental groups of
other polytopes. More precisely, we use the language of words, over the
alphabet of simple transpositions, to obtain conditions that are necessary and
sufficient to characterize the equivalence classes of cycles. The proof
requires only simple combinatorial arguments. As a corollary, we also obtain a
combinatorial proof of the fact that the first Betti number of the complement
of the 3-equal arrangement is equal to $2^{n-3}(n^2-5n+8)-1.$ This formula was
originally obtained by Bj\"orner and Welker in 1995.
