Prove that permutation matrices can be written as the product of transpositions.
So geometrically, we can represent transpositions as lines between two "poles" or sticks. And completing the cycle defines a permutation?
for any $\displaystyle \alpha, \beta \in S_n,$ let $\displaystyle P_{\alpha}, P_{\beta}$ be the corresponding permutation matrices. then $\displaystyle P_{\alpha}P_{\beta}=P_{\beta \alpha }.$ now use this fact that every $\displaystyle \sigma \in S_n$ is a product of transpositions.