Let $n\in\mathbb{N}$, let $\sigma\in S_n$, and let $m\in\mathbb{Z}_{\ge 0}$. Suppose that $\sigma$ admits a [transposition decomposition](/theorems/777) of length $m$, meaning that there exist transpositions $\tau_1,\dots,\tau_m\in S_n$ such that
Consequently, if the same permutation $\sigma$ is written as a product of transpositions in two different ways, then the two lengths have the same parity.