Order of Product of Disjoint Permutations/Examples

Examples of Order of Product of Disjoint Permutations

Permutations in $S_9$

Consider the permutation given in cycle notation as

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

Its order is given by:

$\order \rho = 12$

Non-Disjoint Permutations in $S_9$

Consider the permutation given in cycle notation as

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

Its order is given by:

$\order \rho = 7$

and not $\lcm \set {4, 3, 2} = 12$.