Past exam of the mathematics course of the University of Oxford / 2014 / Prelims / Mathematics I / 6 / b / i

Conjugation of a permutation relabels each symbol by the inverse conjugating permutation. Specifically,To verify the direction of the inverse, apply the left side to ${\tau }^{-1}(a_j)$: $\tau$ first takes it to $a_j$, the original cycle takes it to $a_{j+1}$, and ${\tau }^{-1}$ takes it to ${\tau }^{-1}(a_{j+1})$, with cyclic indexing. Any symbols fixed by the original permutation are similarly relabelled and remain fixed. More generally, conjugate every cycle of a disjoint-cycle decomposition in this way; disjointness is preserved because ${\tau }^{-1}$ is a bijection.