If two cycles are disjoints, then they commute I'm trying to prove that two $\tau$-cycles commute provided that these cycles are disjoints. Hungerford in his book says the following remark about this fact:

Intuitively clearly this is true, but how can we prove this formally?
Thanks in advance

You can probably do this more formally by considering the permutations as elements which act on a set, namely the set of letters $\{1, \ldots, n\}$. In such a case, a cycle cyclically permutes a sub-collection of letters $\{\ell_1, \ldots, \ell_r\}$, and the fact that two cycles commute tell us that these sets which are permuted are disjoint. It should be then clear that the order in which we perform this permutation is irrelevant.
I should note that it is important that the representation of the group elements as permutations of letters is a faithful representation, otherwise this doesn't work. But this is certainly the case here.
