The $(2,3)$-generation of the finite unitary groups

TL;DR

This paper proves that the special unitary groups $SU_n(q^2)$ are generated by elements of orders 2 and 3 for all sufficiently large n and any prime power q, extending known results in group theory.

Contribution

The paper establishes the $(2,3)$-generation of $SU_n(q^2)$ for all $n \ugeq 8$ and any prime power q, filling gaps in the understanding of finite unitary groups' generation.

Findings

$SU_n(q^2)$ are $(2,3)$-generated for $n \ugeq 8$

Most $SU_n(q^2)$ and $PSU_n(q^2)$ are $(2,3)$-generated except for specific small cases

The results extend the class of known finite groups with $(2,3)$-generation.

Abstract

In this paper we prove that the unitary groups $SU_n(q^2)$ are $(2,3)$-generated for any prime power $q$ and any integer $n\geq 8$. By previous results this implies that, if $n\geq 3$, the groups $SU_n(q^2)$ and $PSU_n(q^2)$ are $(2,3)$-generated, except when $(n,q)\in\{(3,2),(3,3),(3,5),(4,2), (4,3),(5,2)\}$.