Permutation_Induces_Equivalence_Relation

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$. Let $\mathcal R_\pi$ be the relation defined by $i \mathrel {\mathcal R_\pi} j \iff \exists k \in \Z: \map {\pi^k} i = j$ Then $\mathcal R_\pi$ is an equivalence relation.