Exercise 2.4.10 ($Q_8$ is isomorphic to a subgroup of $\mathrm{SL}_3(\mathbb{F}_3)$)

Question

Prove that the subgroup of $\mathrm{SL}_2(\mathbb{F}_3)$ generated by $\begin{pmatrix} 0 & -1 \ 1 & 0\end{pmatrix}$ and $\begin{pmatrix} 1 & 1 \ 1 & -1\end{pmatrix}$ is\ isomorphic to the quaternion group of order $8$. [Use a presentation for $Q_8$.]

Richard Ganaye · Accepted Answer

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\begin{proof} We write $I_2$ the identity element of $\mathrm{SL}_2(\mathbb{F}_3)$, and put
$$I = \begin{pmatrix} 0 & -1 \ 1 & 0\end{pmatrix}, \qquad J = \begin{pmatrix} 1 & 1 \ 1 & -1\end{pmatrix}.$$
Let $Q = \langle I, J angle$. Then $I^2 = J^2 = -I_2 \in Q$.

Put $K = IJ = \begin{pmatrix} -1 & 1 \ 1 & 1\end{pmatrix} \in Q$. Then 
\begin{align*}
Q &\supseteq \{I_2, I , J , K ,-I_2, -I,-J,-K\} \
&= \left\{
\begin{pmatrix} 1 & 0 \ 0 & 1\end{pmatrix},
\begin{pmatrix}0 & -1 \ 1 & 0 \end{pmatrix},
\begin{pmatrix} 1 & 1 \ 1 & -1\end{pmatrix},
\begin{pmatrix} -1 & 1 \ 1 & 1 \end{pmatrix},
\begin{pmatrix} -1 & 0 \ 0 & -1\end{pmatrix},
\begin{pmatrix}0 & 1 \ -1 & 0 \end{pmatrix},
\begin{pmatrix} -1 & -1 \ -1 & 1\end{pmatrix},
\begin{pmatrix} 1 & -1 \ -1 & -1 \end{pmatrix}
ight\}
\end{align*}
Since these $8$ matrices are distinct, this shows that 
$$|Q| =  |\langle I, J angle| \geq 8.$$
By Exercise 6.3.7 (or 1.5.3), we know that 
$$Q_8 \simeq \langle a,b \mid a^2 = b^2, \ a^{-1} b a = b^{-1} angle.$$
We write $i, j$ the generators of $Q_8$.

Note that $I^2 =J^2 = -I_2$ and $I^{-1} J  I = J^{-1} = \begin{pmatrix} -1 & -1 \ -1& 1\end{pmatrix} = -J$.

By the van Dyck's Theorem (see the note of Ex. 2.4.7.), there exists a surjective homomorphism $\varphi: Q_8 	o Q$ such that $\varphi(i) = I,\varphi(j) = J$. Hence $8 = |Q_8| \geq |Q|$, so $|Q| = 8$, and $\varphi$ is an isomorphism.

In conclusion,
$$\left \langle \begin{pmatrix} 0 & -1 \ 1 & 0\end{pmatrix}, \begin{pmatrix} 1 & 1 \ 1 & -1\end{pmatrix} ight angle \simeq Q_8.$$
\end{proof}

Exercise 2.4.10 ($Q_8$ is isomorphic to a subgroup of $\mathrm{SL}_3(\mathbb{F}_3)$)

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Exercise 2.4.10 ($Q_8$ is isomorphic to a subgroup of $\mathrm{SL}_3(\mathbb{F}_3)$)

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