Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 7 / a / i

Conjugating a cycle simply relabels its entries. With permutations composed from right to left,This follows by applying each side to $\rho (a_j)$ and to points outside the displayed cycle. Consequently conjugation preserves the lengths in the disjoint cycle decomposition, including fixed points, and therefore preserves cycle type.

Conversely, suppose two permutations have the same cycle type. Pair their cycles of equal length, including their one-cycles. Define a bijection $\rho$ by mapping the entries of each cycle of the first permutation, in their cyclic order, to the entries of the paired cycle of the second. Because the cycles partition $\{1,\dots ,n\}$, these assignments combine into a permutation. The displayed identity then conjugates the first permutation to the second. Thus they are conjugate group elements exactly when their cycle types agree.