<s>[INST] Generators of $PSL(3,2)$ Is it true that a set of generators for $PSL(3,2)\simeq SL(3,2)$ is:
$$\alpha=\left(\begin{array}{ccccccc}
0&0&1\\
0&1&0\\
1&0&0\\
\end{array}\right),$$
$$\beta=\left(\begin{array}{ccccccc}
1&0&0\\
0&0&1\\
0&1&0\\
\end{array}\right),$$
$$\gamma=\left(\begin{array}{ccccccc}
1&0&0\\
0&1&0\\
0&1&1\\
\end{array}\right)?$$
How can I check/prove this fact? Thanks!
 [/INST] Start with any matrix $A\in PSL(3,2)$ and use elementary row
operations (Gaussian elimination) to reduce $A$ to the identity matrix. Each elementary
operation can be accomplished by multiplying on the left by one of the following nine elementary matrices: 
$$
E_{12}=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix},\;
E_{13}=\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix},\;
E_{23}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{pmatrix},\;
$$
$$
E_{21}=\begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix},\;
E_{31}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{pmatrix},\;
E_{32}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 1\end{pmatrix},\;
$$
$$
R_{12}=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix},\;
R_{13}=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{pmatrix},\;
R_{23}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{pmatrix}.
$$
Now noting that each elementary
matrix equals its own inverse, we see that $PSL(3, 2)$ is generated by the
nine elementary matrices listed above. In fact, many of these matrices are redundant,
and the set $\{R_{13}, R_{23}, E_{32}\}$, or $\{E_{23}, R_{12}, R_{23}\}$ already generates the whole group. This follows from formulas like
$R_{13} = R_{12}R_{23}R_{12}$, $E_{13} = R_{12}E_{23}R_{12}$, etc. Indeed, the minimal number of generators is even equal to two, see here.
</s>