Cycle Notation/Examples/Permutations in S8
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Examples of Cycle Notation
Consider the permutations in $S_8$, presented in two-row notation as:
| \(\ds \pi\) | \(=\) | \(\ds \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 3 & 4 & 5 & 6 & 7 & 2 & 1 & 8 \end{pmatrix}\) | ||||||||||||
| \(\ds \rho\) | \(=\) | \(\ds \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 \end{pmatrix}\) |
These can be expressed in cycle notation as:
| \(\ds \pi\) | \(=\) | \(\ds \begin{pmatrix} 1 & 3 & 5 & 7 \end{pmatrix} \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}\) | ||||||||||||
| \(\ds \rho\) | \(=\) | \(\ds \begin{pmatrix} 1 & 8 \end{pmatrix} \begin{pmatrix} 2 & 7 \end{pmatrix} \begin{pmatrix} 3 & 6 \end{pmatrix} \begin{pmatrix} 4 & 5 \end{pmatrix}\) |
We have that:
| \(\ds \pi \rho\) | \(=\) | \(\ds \begin{pmatrix} 1 & 3 & 5 & 7 \end{pmatrix} \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} \begin{pmatrix} 1 & 8 \end{pmatrix} \begin{pmatrix} 2 & 7 \end{pmatrix} \begin{pmatrix} 3 & 6 \end{pmatrix} \begin{pmatrix} 4 & 5 \end{pmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{pmatrix} 1 & 8 & 3 & 2 \end{pmatrix} \begin{pmatrix} 4 & 7 \end{pmatrix} \begin{pmatrix} 5 & 6 \end{pmatrix}\) |
| \(\ds \rho \pi\) | \(=\) | \(\ds \begin{pmatrix} 1 & 8 \end{pmatrix} \begin{pmatrix} 2 & 7 \end{pmatrix} \begin{pmatrix} 3 & 6 \end{pmatrix} \begin{pmatrix} 4 & 5 \end{pmatrix} \begin{pmatrix} 1 & 3 & 5 & 7 \end{pmatrix} \begin{pmatrix} 2 & 4 & 6 \end{pmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{pmatrix} 1 & 6 & 7 & 8 \end{pmatrix} \begin{pmatrix} 2 & 5 \end{pmatrix} \begin{pmatrix} 3 & 4 \end{pmatrix}\) |
| \(\ds \pi^2 \rho\) | \(=\) | \(\ds \pi \paren {\pi \rho}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{pmatrix} 1 & 3 & 5 & 7 \end{pmatrix} \begin{pmatrix} 2 & 4 & 6 \end{pmatrix} \begin{pmatrix} 1 & 8 & 3 & 2 \end{pmatrix} \begin{pmatrix} 4 & 7 \end{pmatrix} \begin{pmatrix} 5 & 6 \end{pmatrix}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \begin{pmatrix} 1 & 8 & 5 & 2 & 3 & 4 \end{pmatrix} \begin{pmatrix} 6 & 7 \end{pmatrix}\) |
$\pi$ is of order $\lcm {4, 3} = 12$, and is of odd parity
$\rho$ is of order $2$, and is of even parity
$\pi \rho$ and $\rho \pi$ are both of order $\lcm {4, 2} = 4$, and are of odd parity
$\pi^2 \rho$ is of order $\lcm {6, 2} = 6$, and is of even parity.
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Exercise $1$