Order of Product of Disjoint Permutations/Examples/Permutations in S9

Example of Order of Product of Disjoint Permutations

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:

$\order \rho = 12$

Proof

$\rho$ is the product of $3$ disjoint permutations, of orders $4$, $3$ and $2$.

From Order of Product of Disjoint Permutations:

$\order \rho = \lcm \set {4, 3, 2} = 12$

$\blacksquare$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $9$: Permutations: Example $9.9$