Rank of a special linear group over a finite field What is the rank (minimal number of group generators) of $SL(n,\mathbb{F})$ in the situation when $SL(n,\mathbb{F})$ is not perfect (i.e. when $SL(n,\mathbb{F})$ is different from $SL(2,\mathbb{F}_2)$ and $SL(2,\mathbb{F}_3)$)? If the general formula is not known, are there $n$ and $F$ which make $rank(SL(n,\mathbb{F}))>k$ for a fixed $k$ (or at least for $k=2$)?
 A: L.E. Dickson proved that when $\mathbb{F}$ is a finite field of odd characteristic $p$, then 
${\rm SL}(2,\mathbb{F}) = \langle \left(\begin{array}{clcr} 1&1\\0&1\end{array} \right), 
\left(\begin{array}{clcr} 1&0\\\lambda &1\end{array} \right) \rangle$ when $\lambda$ is a generator of the multiplicative group of $\mathbb{F}$, except when $|\mathbb{F}| = 9.$
Also, ( a very special case of a much more general result of) R. Steinberg proved that ${\rm PSL}(n,\mathbb{F})$ is generated by two elements whenever $\mathbb{F}$ is a finite field and ${\rm PSL}(n,\mathbb{F})$ is simple. Since the center of ${\rm SL}(n,\mathbb{F})$ is contained in its Frattini subgroup in that case, we know that ${\rm SL}(n,\mathbb{F})$ is still generated by $2$ elements.
A: For any finite field $F$ and $n \ge 2$, ${\rm SL}(n,F)$ is generated by $A$ and $B$, where $A$ is the diagonal matrix with entries $(\omega,\omega^{-1},1,1,\ldots,1)$, for a primitive field element $\omega$ and
$$B = \left(\begin{array}{ccccccc}-1&0&0&\ldots&0&0&1\\
-1&0&0&\ldots&0&0&0\\
0&-1&0&\ldots&0&0&0\\
0&0&-1&\ldots&0&0&0\\
&&&\ldots&&&&\\
0&0&0&\ldots&0&-1&0\\
\end{array}\right).$$
A: Since $SL(n, \mathbb{Z})$ is generated by two matrices (a result of Hua and Reiner from 1948), the same is true for $SL(n, \mathbb{Z}/p)$ for any prime $p.$
