We know that every finite simple group can be generated by $2$ elements.


This (correct me if I'm wrong) was proved, as far as I know, by Steinberg (Steinberg, R. (1962). Generators for Simple Groups. Canadian Journal of Mathematics, 14, 277-283) for Chevalley groups and by Aschbacher and Guralnick (M. Aschbacher, R. Guralnick, Some applications of the first cohomology group, Journal of Algebra, Volume 90, Issue 2, 1984, Pages 446-460, Theorem B) for the other simple groups, using the classification of the finite simple groups (CFSG).


We also know, by an easy application of Lagrange's theorem, that a group of order $n$ can be generated by at most $\log\_2(n)$ elements.


My question is the following. Are there results proving that every simple group of finite order $n$ can be generated by $d$ elements, where $d$ is some quantity between $2$ and $\log\_2(n)$ (for example $\log\_2 \log\_2 (n)$ or - more optimistically - an unspecified constant) without using the classification of finite simple groups?



CFSG-free bound for the number of generators of a finite simple group
Still elementary, but marginally stronger than the $\log\_{2}(n)$ bound is: every non-Abelian finite simple group $G$ of order $n$ can be generated by fewer than $\log\_{p}(n)$ elements, where $p$ is the largest prime divisor of $\lvert G\rvert$. In particular (using Burnside's $p^{a}q^{b}$-theorem), such a group $G$ can be generated by fewer than $\log\_{5}(n)$ elements, since we always have $p \geq 5$. This improves the $\log\_{2}(n)$ bound by a factor of at least $2$, but is still very weak. The proof is easy and elementary— $G$ is generated by its elements of order $p$. Let $S$ be a set of elements of order $p$ of $G$ which generates $G$ with $\lvert S\rvert$ minimal. Then for each positive integer $j$, any $j$ elements of $S$
generate a subgroup of order $p^{j}$ or greater, so $\lvert S\rvert < \log\_{p}(\lvert G\rvert)$.


I don't know what the best known "elementary" bound is (or even if this is a well-defined quantity). I seriously doubt whether there is a constant-type bound known without using CFSG.


