Question

Find the order of each permutation.

(c)$\left(123\right)\left(435\right)$

Short answer

The order of a permutation$\left(123\right)\left(435\right)$ is 5

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.

Permutation σ2 and σ3

$$\begin{gathered} {\sigma }^2=\sigma ·\sigma \\ =\left(123\right)\left(435\right)·\left(123\right)\left(435\right) \\ =\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 2& 3& 5& 1& 4\end{array}\right).\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 2& 3& 5& 1& 4\end{array}\right) \\ =\left(\begin{array}{ccccc}1& 2& 3& 4& 5\\ 3& 5& 4& 2& 1\end{array}\right) \end{gathered}$$

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

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

Permutationσ4and σ5

Find the permutation ${\sigma }^4$as follows:

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

Find the permutation ${\sigma }^5$as follows:

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

Here, $n=5$is the smallest positive integer such that ${\sigma }^5=\in$

Therefore, the order of a permutation$\left(123\right)\left(435\right)$ is 5.