Even permutation

Let $\alpha$ and $\beta$ be elements of $\mathcal{S}_n$. Prove that ${\alpha}^{-1}{\beta}^{-1}\alpha \beta$ is an even permutation.

Solution

The sign function ($1$ for an even permutation and $-1$ for an odd permutation) is a homomorpism from the symmetric group to a group with two elements. That two-element group is abelian (commutative), so the product of a bunch of elements and their inverses in it, in any order, is the identity. Pull back, and the product of the original elements and their inverses was an even permutation.