Order of Cycle is Length of Cycle

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

Let $\pi \in S_n$ be a cyclic permutation of length $k$.

Then:
 * $\order \pi = k$

where:
 * $\order \pi$ denotes the order of $\pi$ in $S_n$.

Proof
Let $\pi = \tuple {a_0, a_1, \ldots, a_{k - 1} }$.

Observe that:

Hence the result, by definition of order of group element.