Permutation Induces Equivalence Relation/Corollary

Theorem

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

Let $\pi \in S_n$.

Let $\RR_\pi$ be the relation defined as:

$i \mathrel {\RR_\pi} j \iff \exists k \in \Z: \map {\pi^k} i = j$

Then:

$i \mathrel {\RR_\pi} j$ if and only if $i$ 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.

$\blacksquare$