Definition:Cycle Decomposition

Definition

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

Let $\pi$ be a permutation on $S_n$.

The cycle decomposition of $\pi$ is the product of the disjoint cycles in which $\pi$ can be expressed.

Examples

Permutation in $S_7$

Consider the permutation given in two-row notation as:

$\rho = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 6 & 1 & 7 & 3 & 2 & 4 \end{pmatrix}$

The cycle decomposition for $\rho$ is:

$\begin{pmatrix} 1 & 5 & 3 \end{pmatrix} \begin{pmatrix} 2 & 6 \end{pmatrix} \begin{pmatrix} 4 & 7 \end{pmatrix}$

Permutation in $S_9$

Consider the permutation given in two-row notation as:

$\rho = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 1 & 4 & 6 & 2 & 8 & 9 & 7 & 5 & 3 \end{pmatrix}$

The cycle decomposition for $\rho$ is:

$\begin{pmatrix} 1 \end{pmatrix} \begin{pmatrix} 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 5 & 8 \end{pmatrix} \begin{pmatrix} 7 \end{pmatrix}$

or, omitting the $1$-cycles:

$\begin{pmatrix} 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 5 & 8 \end{pmatrix}$

Product of Permutations in $S_9$

Consider the product permutations:

$\rho = \begin{pmatrix} 1 & 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 5 & 7 & 9 \end{pmatrix} \begin{pmatrix} 1 & 3 & 9 \end{pmatrix} \begin{pmatrix} 2 & 3 & 4 & 5 & 6 & 8 \end{pmatrix}$

Then $\rho$ evaluates to:

$\begin{pmatrix} 1 & 5 & 6 & 8 & 4 & 7 & 9 & 2 & 3 \end{pmatrix}$

Also see

• Existence and Uniqueness of Cycle Decomposition

• Definition:Cycle Type

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Example $9.6$