Decomposition of a permutation into disjoint cycles

Statement

Let $X$ be a finite set and let $\sigma \in \operatorname{Sym}(X)$. Then $\sigma$ is a product of cycles of length at least $2$ whose sets of moved points are pairwise disjoint. This product is unique up to the order of the factors.

Remarks

The cycles come from the orbits of $\langle \sigma \rangle$ on $X$: each orbit with more than one point gives one cycle. The order of the factors does not matter, because disjoint cycles commute. For example, the permutation $1 \mapsto 2$, $2 \mapsto 1$, $3 \mapsto 4$, $4 \mapsto 5$, $5 \mapsto 3$ in $S_5$ is $(1\ 2)(3\ 4\ 5)$.

Explanations

L1 human

Start at a point and follow the permutation until you return. This gives one cycle; then start again at a point that you did not visit yet. When all points are visited, the permutation is the product of the cycles that you found.