Question

(a) Show that the group$\mathit{G}\mathit{L}\mathbf{(}\mathbf{2}\mathbf{,}{\mathbf{Z}}_2\mathbf{)}$has order 6 by listing all its elements.

(b) Show by example that the groups and $\mathit{G}\mathit{L}\mathbf{(}\mathbf{2}\mathbf{,}{\mathbf{Z}}_2\mathbf{)}$are non-abelian.

Short answer

(a) It is proved that group$GL(\mathbf{2}\mathbf{,}{Z}_2)$ has order 6.

(b) It is proved that the groups$GL(\mathbf{2}\mathbf{,}ℝ)$ and$GL(\mathbf{2}\mathbf{,}{Z}_2)$ are non-abelian.

Step-by-step solution

Solution to Part (a)

Write the general linear groups of $n\times n$matrices, which is defined over a field .

$F$

$GL(n,F)=\{A\in M_{n\times n}\left(F\right)|\det A\ne 0\}$

Now, list all the elements of$M_{n\times n}\left({\mathbb{Z}}_2\right)$.

localid="1659416805263" $GL(2,{\mathbb{Z}}_2)=\left\{\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]\right\}$

Hence, the given statement is proved.

Solution to Part (b) 

As in the group$GL(2,{\mathbb{Z}}_2)$we have,

localid="1659416915309" $\begin{array}{l}\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\\ \left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\end{array}$

Now,in the group$GL(2,ℝ)$we have,

localid="1659416973104" $\begin{array}{l}\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}2& 1\\ 1& 0\end{array}\right]\\ \left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 1& 1\end{array}\right]\end{array}$

Therefore, the groups are not abelian.

Hence, the given statement isproved.