Question

Find the order of each permutation.

(a) (12)

Short answer

The order of a permutation (12) 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 (12)

Apply the permutation $\sigma$twice then,

Here, 1 is mapped to 2 and 2 is mapped to 1 so that the product maps 1 to 1. Similarly, 2 is mapped to 1 and 1 is mapped to 2 so that the product maps 2 to 2.

Thus, the product $\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right).\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)=\left(\begin{array}{cc}1& 2\\ 1& 2\end{array}\right)$. Hence $n=2$, is the smallest positive integer such that ${\sigma }^2=\in$

Therefore, the order of a permutation (12) is 2.