Question

Find the order of each permutation.

(d) What do you think the order of ( 123456789 ) is?

Short answer

The order of a permutation ( 123456789 ) is 9.

Step-by-step solution

Order of a permutation

The order of a permutation $\sigma$ is the smallest positive integer $n$such that ${\sigma }^n=\in$, where $\in$is the identity permutation.

If $\sigma =\left(a_1{a}_2\cdots a_m\right)$is a cycle of length $m$from $\left\{1,2,\cdots ,n\right\}$then the order $\sigma$ of is$m$

Order of a permutation ( 123456789 )

Here,$\sigma =\left(123456789\right)$ is a cycle of length 9 from$\left\{1,2,\cdots ,n\right\}$ then by the definition the order of$\sigma$ is 9.

Therefore, the order of a permutation$\left(123456789\right)$ is 9.