Special linear group matrix multiplication

Question

I've been trying to figure out how the group $SL(2,3)$ works but I don't really understand it.

Say I have the matrix $B = \begin{pmatrix} 1 &1 \\ 1 & -1 \end{pmatrix}$ then how should I calculate $B^2$. (apparently B has order 2)

Answer 1

I guess that with SL$(2,3)$ you mean SL$_2(\Bbb F_3)$, that is all $2\times 2$ matrices with coefficients in the finite field $\Bbb F_3$. I remind you that in this field $2=-1$.

In such case, yourmatrix $B$ has order $4$. Indeed, as you have already commented above,

$$B^2= \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}= \begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix}= \begin{pmatrix} -1 & 0\\ 0 & -1 \end{pmatrix}$$

Similarly

$$B^3= \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix} \begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix}= \begin{pmatrix} 2 & 2\\ 2 & -2 \end{pmatrix}= \begin{pmatrix} 2 & 2\\ 2 & 1 \end{pmatrix}$$

Finally $$B^4=(B^2)^2= \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}$$

However, the matrix $B$ has order $2$ if you consider PSL$_2(\Bbb F_3)$, defined as

$$ \text{SL}_2(\Bbb F_3)/\{\pm Id\}$$