Question

Find the order of each permutation.

(c) ( 1234 )

Short answer

The order of a permutation ( 1234 ) is 4.

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  ( 1234 )

Apply the permutation $\sigma$twice then,

$$\begin{gathered} {\sigma }^2=\sigma ·\sigma \\ =\left(1234\right)·\left(1234\right) \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\end{array}\right).\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\end{array}\right) \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 4& 1& 2\end{array}\right) \end{gathered}$$

Now, apply the permutation $\sigma$thrice then,

role="math" localid="1651662492493" $$\begin{gathered} {\sigma }^3={\sigma }^2·\sigma \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 3& 4& 1& 2\end{array}\right).\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\end{array}\right) \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 4& 1& 2& 3\end{array}\right) \end{gathered}$$

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

role="math" localid="1651662650795" $$\begin{gathered} {\sigma }^4={\sigma }^3·\sigma \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 4& 1& 2& 3\end{array}\right).\left(\begin{array}{cccc}1& 2& 3& 4\\ 2& 3& 4& 1\end{array}\right) \\ =\left(\begin{array}{cccc}1& 2& 3& 4\\ 1& 2& 3& 4\end{array}\right) \\ =\in \end{gathered}$$

Here, $n=4$is the smallest positive integer such that ${\sigma }^4=\in$

Therefore, the order of a permutation ( 1234 ) is 4.