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