Definition:Cyclic Permutation

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Definition

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

A cyclic permutation of length $k$ is also known as:

a cycle of length $k$

a $k$-cycle

generally, just a cycle.

Not all permutations are cycles.

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

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