<s>[INST] $SL(2, Z_2)$ closed under multiplication How can $SL(2, Z_2)$ (i.e matrices with entries in $\mathbb{Z_2}$ and determinant 1) form a group, when:
$\begin{pmatrix} 1&0 \\ 1&1 \end{pmatrix}$ * $\begin{pmatrix} 1&1 \\ 0&1 \end{pmatrix}$ = $\begin{pmatrix} 1&1 \\ 1&2 \end{pmatrix}$ which doesn't have entries in $\mathbb{Z_2}$
and if we let $\begin{pmatrix} 1&1 \\ 1&2 \end{pmatrix} =$ $\begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix}$ then $\begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix}$ has determinant $-1$ 
So either way we have the matrices are not closed under multiplication. Where am I wrong?
 [/INST] $1=-1$ in $\mathbb Z_2$, and you should reduce all entries modulo $2$, so the first matrix has a zero in the bottom right entry.
i.e: $$\begin{pmatrix}1&1\\1&2 \end{pmatrix}=\begin{pmatrix}1&1\\1&0 \end{pmatrix}$$
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