Transpositions_of_Adjacent_Elements_generate_Symmetric_Group

Let $n \in \Z: n > 1$. Let $S_n$ denote the symmetric group on $n$ letters. Then the transpositions $a_k = \begin{pmatrix} k & k + 1 \end{pmatrix}$ for $1 \le k < n$ are a set of generators for $S_n$. They satisfy the relations {{eqn | l = a_k^2 | r = e | c = (for $1 \le k < n$) }} {{eqn | l = \paren {a_k a_{k + 1} }^3 | r = e | c = (for $1 \le k < n - 1$) }} {{eqn | l = \paren {a_i a_j}^2 | r = e | c = (for $1 \le i, j < n, \size {i - j} > 1$) }}