The question is in the title. If $\Gamma\subset\mathrm{GL}(\Bbb R^d)$ is a *finite* matrix group, can it be generated by (at most) $d$ elements?


I suspect that this hope is too naive, but I have no counterexamples. I would also be interested in bounds on the number of generators.



Is every finite $d$-dimensional matrix group generated by $d$ elements?
The answer to the question is yes.


In Theorem 1.2 of [this paper](https://www.sciencedirect.com/science/article/pii/S0021869313002160). the authors (Colva Roney-Dougal and myself) prove that if $G \le {\rm GL}(n,F)$ with $F$ a field, where


(a) $G$ is finite;


(b) either $G$ is completely reducible or ${\rm char} F = p$ and $O\_p(G)=1$; and


(c) $F$ does not contain a primitive fourth root of unity,


then $d(G) \le n$.


This is best possible when ${\rm char} F \ne 2$, because there is an obvious embedding of $C\_2^n$ in ${\rm GL}(n,F)$.


