Question

Prove that $\left\{\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}\right)\right\}$is a subgroup of${\mathbf{S}}_{\mathbf{3}}$ but not normal.

Short answer

It is proved that$\left\{\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right)\right\}$ is a subgroup of${\mathrm{S}}_3$ but not normal.

Step-by-step solution

Prove that given set is subgroup 

Assume that the given subset is$\mathrm{H}=\left\{\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}\right)\right\}$ .

Prove H is a subgroup as:

$\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right)\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right)=\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}\right)$

Hence, it is proved that$\mathrm{H}$ is a subgroup.

Prove that H is not normal

Consider the element $\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 2\end{array}\right)\in {\mathrm{S}}_3$, this implies that:

$\begin{array}{rcl}\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 2\end{array}\right)\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right)\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 2\end{array}\right)& =& \left(\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}\right)\\ & \in & \mathrm{H}\end{array}$

This implies that $\mathrm{H}$is not normal.

Hence, it is proved that $\left\{\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}\right)\right\}$is a subgroup of${\mathrm{S}}_3$ but not normal.