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.

Cycle Type
Let $$\pi, \rho \in S_n$$.

Then $$\pi$$ and $$\rho$$ have the same cycle type if they have the same number of cycles of equal length.

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.