On the (2,3)-generation of the finite symplectic groups

TL;DR

This paper proves that symplectic groups over finite fields are generated by elements of orders 2 and 3 for all sufficiently large dimensions, advancing the classification of (2,3)-generated finite simple groups.

Contribution

It establishes that all symplectic groups $Sp_{2n}(q)$ with $n geq 2$ are (2,3)-generated, filling a significant gap in the classification of such groups.

Findings

$Sp_{2n}(q)$ are (2,3)-generated for all $n geq 2$

Implication that $PSp_{2n}(q)$ are (2,3)-generated for all $n geq 1$ except specific cases

Advances the classification of finite simple groups by (2,3)-generation

Abstract

This paper is a new important step towards the complete classification of the finite simple groups which are $(2, 3)$-generated. In fact, we prove that the symplectic groups $Sp_{2n}(q)$ are $(2,3)$-generated for all $n\geq 4$. Because of the existing literature, this result implies that the groups $PSp_{2n}(q)$ are $(2,3)$-generated for all $n\geq 2$, with the exception of $PSp_4(2^f)$ and $PSp_4(3^f)$.