Existence and Uniqueness of Cycle Decomposition

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Every element of $S_n$ may be uniquely expressed as a product of disjoint cycles, up to the order of factors.

This expression is known as the cycle decomposition of the permutation.

Proof
Let $\pi \in S_n$ be a permutation on $S_n$.

Let $\mathcal R_\pi$ be the equivalence defined in Permutation Induces Equivalence Relation.

Then the equivalence classes induced by $\mathcal R_\pi$ are the required cycles.

The uniqueness follows from the fact that the partition of the permutation into $\mathcal R_\pi$-classes can be done in only one way.

Also see

 * Cycle type