# Cycle Notation/Examples/Permutations in S8

## 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.