# On $(2,3)$-generation of finite simple classical groups

A group $$G$$ is called $$(a,b)$$-generated if $$G=\langle x,y\rangle$$ for some $$x,y\in G$$ with $$|x|=a$$ and $$|y|=b$$.

I know some of the histories on this problem. For example, in this early paper in 1996 Liebeck and Shalev proved that

Theorem. All but finitely many finite simple classical groups other than $$\operatorname{PSp}_4(2^f)$$ or $$\operatorname{PSp}_4(3^f)$$ are $$(2,3)$$-generated.

In this paper in 2017, King proved that

Theorem. Every finite simple group is $$(2,r)$$-generated for some prime $$r\ge 3$$.

Is there any other result on $$(2,3)$$-generation or $$(2,r)$$-generation of finite simple classical groups? For example, the $$(2,3)$$-generation for low-dimensional classical groups? Or is there any lower bound (w.r.t the dimension and the order of field) of $$(2,3)$$-generation?

• Liebeck and Shalev proved a later result that all sufficiently large-rank classical groups are $(r,s)$-generated for any two primes, at least one of which is odd. I have proved, but never got round to publishing, that all low-rank exceptional groups (up to $F_4(q)$) are $(2,p)$-generated for every odd prime $p$ dividing their order. Commented Jul 11, 2020 at 19:01
• An interesting example is provided by ${\rm PSU}(3,3)$. This is the only non-Abelian finite simple group which can not be generated by three involutions. Consequently, ${\rm PSU}(3,3)$ is not $(2,3)$-generated, for it were generated by $a$ and $b$ with $a^{2} = b^{3} = 1$, then it would still be generated by $\{a,a^{b},a^{b^{2}}\}.$ Commented Jul 12, 2020 at 14:47

This is not a definitive answer (I doubt there is one), but too long for a comment.

Indeed, it is known that among the finite simple groups there are, apart from $$\operatorname{PSp}_4(2^k)$$, $$\operatorname{PSp}_4(3^k)$$ and $${}^2\mathsf{B}_2(2^{2k+1})$$, only finitely many which are not (2,3)-generated. The case of classical groups was done by Liebeck and Shalev, while Lübeck and Malle dealt with the exceptional groups. The complete list of non-(2,3)-generated finite (quasi-)simple groups is not known at the moment, and it takes some effort to show for a particular group of Lie type that it is not (2,3)-generated, see, for example, More classical groups which are not (2, 3)-generated by M. Vsemirnov. It is shown there that $$\operatorname{PSU}_5(4)$$ is not (2,3)-generated, as well as $$\operatorname{Sp}_6(q)$$ for any odd $$q=p^k$$.

Another possible direction is the study of (2,3,7)-generation, that is, whether there is a generating pair of an involution $$x$$ and an element $$y$$ of order 3, such that $$xy$$ is of order 7, see Hurwitz Groups and Hurwitz Generation by Tamburini and Vsemirnov.

• You need to be a little careful when you start mentioning quasi-simple groups. For example, ${\rm SL}(2,q)$ ($q$ odd) is never $(2,3)$-generated, as its unique involution is central. Commented Jul 11, 2020 at 9:32