Uniqueness of factorising permutations into cycles

Question

Consider the product of cycles $\pi = \pi_1 \cdot \pi_2 \cdot \ldots \cdot \pi_r = \mu_1 \cdot \mu_2 \cdot \ldots \cdot \mu_s$. Suppose $\pi_i = (i_1, i_2, \ldots)$. Since the cycles are disjoint(if the cycles weren’t disjoint we’d get more than two cycles with the same element), there’s a unique cycle, say, $\mu_i = (j_1, j_2, \ldots)$ such that that $i_1 = j_1$ (any element of a cycle can be moved up to the initial position if need be). Then $i_2 = \pi(i_1) = \mu(j_1) = i_2$. Similarly, $i_3 = j_3$. The other cycles are dealt with in the same way.

Does the above make sense? Is it important here to mention that cycles commute?

Answer 1

Seems fine to me, no need to state that disjoint cycles commute (which is easy to prove anyway).

I would only emphasize more on the fact that it is sufficient to prove that any cycle in $\{\pi_1, ..., \pi_r\}$ is in $\{\mu_1, ..., \mu_r\}$ because both sets play a symmetrical role.