Definition:Cyclic Permutation

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

Let $$\rho \in S_n$$.

Then $$\rho$$ is a $$k$$-cycle if there exists $$k \in \Z: k > 0$$ and $$i \in \Z$$ such that:
 * 1) $$k$$ is the smallest such that $$\rho^k \left({i}\right) = i$$;
 * 2) $$\rho$$ fixes each $$j$$ not in $$\left\{{i, \rho \left({i}\right), \ldots, \rho^{k-1} \left({i}\right)}\right\}$$.

A $$k$$-cycle is alternatively referred to as a cycle of length $$k$$, or, generally, just a cycle.

The $$k$$-cycle $$\rho$$ is usually denoted $$\begin{bmatrix} i & \rho \left({i}\right) & \ldots & \rho^{k-1} \left({i}\right) \end{bmatrix}$$ (see Cycle Notation).

Comment
Not all permutations are cycles.

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

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