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

Let \(\displaystyle{\sigma:S_{n}\to \mathbb{Z}_{2}}\) be the sign map, which is ring epimorphism.

Then,

\(\displaystyle{\begin{aligned} \sigma(a^{-1}\,\beta^{-1}\,a\,\beta)&=-\sigma(a)-\sigma(b)+\sigma(a)+\sigma(b)\\&=0\end{aligned}}\)

so, the permutation \(\displaystyle{a^{-1}\,\beta\,a\,\beta}\) is even.