Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 5 / b / ii / Solution

With products acting from right to left, a permutation cycle satisfiesThe right side sends $a_1$ to $a_2$, each $a_j$ to $a_{j+1}$ for $2\le j<k$, and $a_k$ to $a_1$, while fixing all other symbols. This verifies the formula. Applying it to every cycle in the disjoint cycle decomposition expresses any permutation as a product of transpositions. The identity is represented by the empty product.