The cycle structure of a permutation σ is defined as the unordered list of the sizes of the cycles in the cycle decomposition σ. For example, the permutation σ=(12)(345)(78)(9) has cycle structure (2,3,2,1) which can also be written as (1,2,2,3).
Show that any two permutations α,β∈Sn have the same cycle structure if and only if there exists a permutation γ such that β=γαγ−1. If β=γαγ−1 for some γ∈Sn, then α and β are conjugate.