Question

Show that $GL(2,{\square }_2)$ is isomorphic to S₃by writing out the operation table for each groups

Short answer

It is proved that the group$GL(2,{\square }_2)$ is isomorphic to S₃ .

Step-by-step solution

Elements in GL(2,□2) and S3

The elements in $GL(2,{\square }_2)$are $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right),\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right),\left(\begin{array}{cc}1& 1\\ 1& 0\end{array}\right),\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$and

the elements in S₃are$\left(\begin{array}{ccc}1& 2& 3\\ 1& 2& 3\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 2& 1& 3\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 3& 1& 2\end{array}\right),\left(\begin{array}{ccc}1& 2& 3\\ 1& 3& 2\end{array}\right)$

Operation table forGL(2,□2)

The operation table for $GL(2,{\square }_2)$is shown below

Operation table for S3

The operation table for S₃ is shown below:

The corresponding elements in the operation table of group$GL(2,{\square }_2)$ and S₃are identical.

Therefore, the group $GL(2,{\square }_2)$is isomorphic to S₃.