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

For each symbol $a$, iterate the permutation $\sigma$. Since the symbol set is finite, some iterate repeats. Applying an inverse power of $\sigma$ shows that the first return is to $a$ itself, givingwith the displayed symbols distinct. This is a permutation cycle. If two such lists share a symbol, applying powers of $\sigma$ shows that they contain exactly the same symbols. Hence these group orbits form a partition of a set $\{1,\dots ,n\}$. On each group orbit, $\sigma$ acts as the corresponding permutation cycle, and these disjoint permutation cycles multiply to $\sigma$.

The disjoint cycle decomposition is unique up to reordering the cycles and cyclically rotating the entries within each cycle, with fixed points optionally omitted. Indeed, each cycle must consist of one of the uniquely determined group orbits, and its successive entries must follow the action of $\sigma$.