Find the cardinality of the image of $\varphi.$

Question

Consider the group homomorphism $\varphi : SL_2 (\Bbb Z) \longrightarrow SL_2 (\Bbb Z/ 3 \Bbb Z)$ defined by $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} \overline {a} & \overline {b} \\ \overline {c} & \overline {d} \end{pmatrix}.$$

What is the cardinality of the image of $\varphi$?

Since $\text {Im} (\varphi)$ is a subgroup of $SL_2 (\Bbb Z/ 3 \Bbb Z)$ so by Lagrange's theorem $\#\ \text {Im} (\varphi)\ \big |\ \#\ SL_2 (\Bbb Z/ 3 \Bbb Z) = 24.$ What I have observed is that $\#\ \text {Im} (\varphi) \geq 7.$ So $\#\ \text {Im} (\varphi) = 8,12\ \text {or}\ 24.$

I have just seen that the image contains at least $10$ elements. So the possibility for cardinality of $\text {Im} (\varphi)$ is $12$ or $24.$

Answer 1

The site didn't allow me to post this as a comment, so here is it as an answer.

See lemma 2.2 in this paper: https://arxiv.org/pdf/1306.2385.pdf I found that quite accessible and readable.

Answer 2

For any $n\geq 1$ and any prime number $p$, the canonical map $\pi:SL_n(\mathbb{Z})\to SL_n(\mathbb{F}_p)$ is surjective.

Indeed, it is a standard fact that if $R$ is an Euclidean ring (for example $R=\mathbb{Z}$ or $R=\mathbb{F}_p$), then $SL_n(R)$ is generated by the transvection matrices $$T_{i,j}(a)=I_n+aE_{ij}, i\neq j, a\in R$$ ($E_{ij}$ is the matrix whose entries are $0$, except the one at position $(i,j)$, which is $1$).

So, we just have to check that the image contains any transvection matrix. But this is clear, since $\pi (T_{i,j}(a))=T_{i,j}(\bar{a})$ for all $a\in\mathbb{Z}$.

Hence $f$ is surjective.