Definition:Cyclic Permutation

Definition
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 $k$, if there exists $k \in \Z: k > 0$ and $i \in \Z$ such that:
 * $(1): \quad k$ is the smallest such that $\rho^k \left({i}\right) = i$


 * $(2): \quad \rho$ fixes each $j$ not in $\left\{{i, \rho \left({i}\right), \ldots, \rho^{k-1} \left({i}\right)}\right\}$.

$\rho$ is usually denoted using cycle notation as:
 * $\begin{bmatrix} i & \rho \left({i}\right) & \ldots & \rho^{k-1} \left({i}\right) \end{bmatrix}$

Also known as
A cyclic permutation of length $k$ is also known as:
 * a cycle of length $k$
 * a $k$-cycle
 * generally, just a cycle.

Comment
Not all permutations are cycles.

Here is an example (written in two-row notation) of a permutation which is not a cycle:


 * $\begin{bmatrix}

1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}$