This implication follows from the conjugation formulas.

We write both g and h as a product of disjoint cycles s_i and t_i, respectively, all of length at least 2. Since g and h have the same cycle structure, we can write g = s₁ ··· s_m and h = t₁ ··· t_m in such a way that s_i and t_i have equal length for all i. Suppose s_i = (s_i1, s_i2, ..., s_ipi) and t_i = (t_i1, ..., t_ipi). Denote by k a permutation with k(t_ij) = s_ij for all i from 1 to m and j from 1 to p_i. This is possible since the supports of the s_i are disjoint as well as the supports of the t_i. (Notice that there may be more than one permutation k satisfying these requirements.) The conjugation formulas yield that khk^-1 = g.