Can a product of disjoint cycles be written as a single cycle?

Question

Given a product of disjoint cycles $\in S_n$, can this be written as a single cycle? If yes ever, under what conditions? If never, why?

Answer 1

If you have a product of disjoint cycles $\sigma$ then the orbits under the action of $\langle \sigma \rangle$ correspond exactly to the cycles in the product (including the trivial ones). So the elements within the cycles depends on $\sigma$, and not on the factorization.

Answer 2

The decomposition of a permutation as a product of disjoint cycles of length $\ge 2$ is unique, up to the order of the cycles.

Suppose on the contrary that one has $\;\gamma_1\dotsm\gamma_r=\gamma'_1\dotsm\gamma'_s$. As the inverse of a cycle is a cycle, this comes down to proving that if $\gamma_1\dotsm\gamma_t=e$, then $t=0$. Since the order of a product of disjoint cycles is the l.c.m. of the orders (= lengths) $\ell_i$ of the cycles. However if $t\ge 1$, this l.c.m. is at least $2$, so the product can't be the empty cycle.