Let $G=GL(2,\mathbb{F}_3)$. Prove that $H/Z(G) \cong A_4$, where $Z(G)$ is the center of $G$ and $H=\{A \in G \mid \det (A)=1\}=\ker (\det)$.
We know that $|H|=\dfrac{|G|}{|\ker(\det)|}=\dfrac{48}{2}=24$, and $Z(G)=\left\{\lambda I \mid \lambda \in \mathbb{F}_{3}^{*}\right\}=\left\{\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 2 & 0 \\ 0 & 2 \end{array}\right)\right\}$.
So we have that $|H/Z(G)|=|A_4|$. At this point, if we find an injective homomorphism between $H/Z(G)$ and $A_4$, we can conclude that is an isomorphism. The problem if I don't know how $H$ is made, and consequently how the quotient is made. So, how can I build this homomorphism?
