Permutation Induces Equivalence Relation/Corollary

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

Let $\pi \in S_n$.

Let $\mathcal R_\pi$ be the relation defined as:


 * $i \mathrel{\mathcal R_\pi} j \iff \exists k \in \Z: \pi^k \left({i}\right) = j$

Then:
 * $i \mathrel{\mathcal R_\pi} j$ $i$ and $j$ are in the same cycle of $\pi$.

Proof
We have that Permutation Induces Equivalence Relation.

The equivalence classes of that equivalence relation are the cycles of $\pi$.

Hence the result by definition of equivalence class.