I'm trying to prove that in a symmetric group two disjoint cycles commute. But I suspect that something is not right about my proof (a sense of vagueness). Some hints would be appreciated.
Here's my proof:
Let $\sigma=(s_1 s_2 ... s_n)$, $\tau=(t_1 t_2 ... t_m)$ for some integers $m, n$. That is, $s_i \neq t_j$ for any $i \in [1, n]$, $j \in [1, m]$. Now, $\sigma\tau=(s_1 ... s_n)(t_1 ... t_m)$ is a cycle of disjoint permutations, which cannot be represented by any other disjoint permutations. Thus $\sigma\tau=\tau\sigma$.