Question

Find the order of each permutation.

(a)$\left(13\right)\left(24\right)$

Short answer

The order of a permutation $\left(13\right)\left(24\right)$is 2.

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.

Order of a permutation (13)(24)

Apply the permutation $\sigma$twice. Then,

$$\begin{gathered} {\sigma }^2=\sigma ·\sigma \\ =\left(13\right)\left(24\right)·\left(13\right)\left(24\right) \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 4& 1& 2\end{array}\right).\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 4& 1& 2\end{array}\right) \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 2& 3& 4\end{array}\right) \end{gathered}$$

Here, $n=2$is the smallest positive integer such that${\sigma }^2=\in$

Therefore, the order of a permutation $\left(13\right)\left(24\right)$is 2.