Cycle Decomposition/Examples/Permutation in S9

Example of Cycle Decomposition

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:

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

Sources

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