Let $k \in \Bbb{N}$ and $\sigma \in S_n$. Then there exists $x,y \in \{1,...,n\}$ such that $\sigma^{k} (x,y) \sigma^{-k} = (1,2)$.
Okay. This is rather trivial, if you recall that $\tau (x_1,x_2,..,x_\ell)\tau^{-1} = (\tau(x_1),...,\tau(x_\ell))$, since the above can be written as $(x,y) = \sigma^{-k}(1,2) \sigma^k$ or $(x,y) = (\sigma^{-k}(1),\sigma^{-k}(2))$, and we may take $x = \sigma^{-k}(1)$ and $y = \sigma^{-k}(2)$. What worries me is that this works for all natural numbers $k$ and all permutations $\sigma$. This result seems quirky; is it true, or did I make a mistake?