Question

Find the order of the given element.

(b) $\mathbf{\left(}\begin{array}{ccccccc}\mathbf{1}& \mathbf{2}& \mathbf{3}& \mathbf{4}& \mathbf{5}& \mathbf{6}& \mathbf{7}\\ \mathbf{2}& \mathbf{3}& \mathbf{7}& \mathbf{5}& \mathbf{1}& \mathbf{4}& \mathbf{6}\end{array}\mathbf{\right)}$in ${\mathit{S}}_{\mathbf{7}}$.

Short answer

The, order of $\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)$is 7.

Step-by-step solution

Order of an element

Order of an element$\mathit{a}\mathbf{\in }\mathit{G}$ is defined as a positive integer n such that${\mathit{a}}^{\mathbf{n}}\mathbf{=}\mathit{e}$, where $\mathit{e}$is the identity element in $\mathit{G}$.

Order of the given element

The given permutation is$\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)$.Now,

$\begin{array}{l}{\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)}^2=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 3& 7& 6& 1& 2& 5& 4\end{array}\right)\\ \Rightarrow {\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)}^3=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 7& 6& 4& 2& 3& 1& 5\end{array}\right)\\ \Rightarrow {\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)}^4=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 6& 4& 5& 3& 7& 2& 1\end{array}\right)\end{array}$

$\begin{array}{l}\Rightarrow {\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)}^5=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 4& 5& 1& 7& 6& 3& 2\end{array}\right)\\ \Rightarrow {\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)}^6=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 5& 1& 2& 6& 4& 7& 3\end{array}\right)\\ \Rightarrow {\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)}^7=\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 1& 2& 3& 4& 5& 6& 7\end{array}\right)\end{array}$

Therefore, by definition,

The order of$\left(\begin{array}{ccccccc}1& 2& 3& 4& 5& 6& 7\\ 2& 3& 7& 5& 1& 4& 6\end{array}\right)$is 7.