\begin{definition}[Definition:Cyclic Permutation]
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$ be a permutation on $S$.
Then $\rho$ is a '''cyclic permutation of length $k$''' {{iff}} there exists $k \in \Z: k > 0$ and $i \in \Z$ such that:
:$(1): \quad k$ is the smallest such that $\map {\rho^k} i = i$
:$(2): \quad \rho$ fixes each $j$ not in $\set {i, \map \rho i, \ldots, \map {\rho^{k - 1} } i}$.
$\rho$ is usually denoted using cycle notation as:
:$\begin{pmatrix} i & \map \rho i & \ldots & \map {\rho^{k - 1} } i \end{pmatrix}$
but some sources introduce it using two-row notation:
:$\begin{pmatrix} a_1 & a_2 & \cdots & a_k & \cdots & i & \cdots \\ a_2 & a_3 & \cdots & a_1 & \cdots & i & \cdots \end{pmatrix}$
\end{definition}