Consider the invertible matrices in $\mathbb F\_2^{n\times n}$ which are a multiplicative group structure. Is there a finite set of $2k$ (at a $k\in\mathbb Z\_{\geq1}$ independent of $n$) generators for the group on the condition if $a$ is a generator $a\neq a^{-1}$ and $a^{-1}$ is a member of the set and is considered a different element compared to element $a$ (essentially the smallest set of no elements of order $2$ and is there an explicit example)?



Generators of $SL(n,\mathbb F_2)$?
OK, I'll answer the intended question! ${\rm SL}(2,2) \cong S\_3$ and the answer to the question is no in that case, so assume that $n>2$.


Don Taylor wrote down explicit sets of two generators for the finite classical groups. Search for "D.E. Taylor Pairs of Generators for Matrix Groups". (These are the generators used by the software packages GAP and Magma.)


The generators for ${\rm SL}(n,2)$ are $x = I+E\_{21}$ (a transvection with $1$ in the $(2,1)$ position), and a permutation matrix $y$ for the cycle $(1,2,3,\ldots,n)$.


Now $x$ has order $2$, but $xy$ and $y$ do not (proof left as exercise), so for our inverse-closed generating set we can take $\{y,y^{-1},xy,(xy)^{-1}\}$.


