Question

Find the order of each permutation.

(b)$\left(123\right)\left(456\right)$

Short answer

The order of a permutation$\left(123\right)\left(456\right)$ is 3.

Step-by-step solution

Order of a permutation

Theorder 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 (123)(456)

Apply the permutation $\sigma$twice. Then,

$$\begin{gathered} {\sigma }^2=\sigma ·\sigma \\ =\left(123\right)\left(456\right)·\left(123\right)\left(456\right) \\ =\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 3& 1& 5& 6& 4\end{array}\right).\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 3& 1& 5& 6& 4\end{array}\right) \\ =\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 1& 2& 6& 4& 5\end{array}\right) \end{gathered}$$

Now, apply the permutation $\sigma$thrice. Then,

$$\begin{gathered} {\sigma }^3={\sigma }^2·\sigma \\ =\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 3& 1& 2& 6& 4& 5\end{array}\right).\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 2& 3& 1& 5& 6& 4\end{array}\right) \\ =\left(\begin{array}{cccccc}1& 2& 3& 4& 5& 6\\ 1& 2& 3& 4& 5& 6\end{array}\right) \end{gathered}$$

Here, $n=3$is the smallest positive integer such that ${\sigma }^3=\in$

Therefore, the order of a permutation$\left(123\right)\left(456\right)$ is 3.