Exercise 1.4.2 (Order of the elements of $\mathrm{GL}_2(\mathbb{F}_2)$)

Question

Write out all elements of $\mathrm{GL}_2(\mathbb{F}_2)$ and compute the order of each element.

Richard Ganaye · Accepted Answer

ewcommand{\F}{\mathbb{F}}

\begin{proof} 
We obtained the complete list of the elements of $\mathrm{GL}_2(\F_2)$ in Exercise 1:
$$\mathrm{GL}_2(\F_2) =
 \left \{ 
 \begin{pmatrix} 1 & 0\ 0 & 1\end{pmatrix},
 \begin{pmatrix} 1 & 0\ 1 & 1\end{pmatrix},
 \begin{pmatrix} 0 & 1\ 1 & 0\end{pmatrix},
 \begin{pmatrix} 0& 1\ 1 & 1\end{pmatrix},
 \begin{pmatrix} 1&1\ 0 & 1\end{pmatrix},
 \begin{pmatrix} 1 & 1\ 1 & 0\end{pmatrix}
 ight\}.
$$
Since
$$ \begin{pmatrix} 1 & 0\ 1 & 1\end{pmatrix}^2 =  \begin{pmatrix} 0 & 1\ 1 & 0\end{pmatrix}^2 =  \begin{pmatrix} 1& 1\ 0 & 1\end{pmatrix}^2 = I_2,$$
and 
$$  \begin{pmatrix} 0& 1\ 1 & 1\end{pmatrix}^3 =  \begin{pmatrix} 1 & 1\ 1 & 0\end{pmatrix}^3 = I_2,$$
we obtain the order of each element in $\mathrm{GL}_2(\F_2)$:
\begin{center}
\begin{tabular}{|c|c|}
\hline
$M$ & $|M|$     \
\hline
$\begin{pmatrix} 1 & 0\ 0 & 1\end{pmatrix}$ & 1 \
$ \begin{pmatrix} 1 & 0\ 1 & 1\end{pmatrix}$ & 2\
 $\begin{pmatrix} 0 & 1\ 1 & 0\end{pmatrix}$& 2\ 
$ \begin{pmatrix} 0& 1\ 1 & 1\end{pmatrix}$& 3\
 $\begin{pmatrix} 1&1\ 0 & 1\end{pmatrix}$& 2\
 $\begin{pmatrix} 1 & 1\ 1 & 0\end{pmatrix}$&3\
\hline
\end{tabular}
\end{center}
\end{proof}


$M$	$\| M \|$

$(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$	1
$(\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix})$	2
$(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$	2
$(\begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix})$	3
$(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})$	2
$(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix})$	3

Exercise 1.4.2 (Order of the elements of $\mathrm{GL}_2(\mathbb{F}_2)$)

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Exercise 1.4.2 (Order of the elements of $\mathrm{GL}_2(\mathbb{F}_2)$)

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