The simple classical groups of dimension less than 6 which are (2,3)-generated

TL;DR

This paper classifies certain small-dimensional classical simple groups that are generated by elements of orders 2 and 3, providing explicit generators and extending known results for dimensions 3 and 5.

Contribution

It determines which classical simple groups of dimensions 3 and 5 are (2,3)-generated and supplies explicit generators, expanding the classification beyond previously known cases.

Findings

PSL_3(q) are (2,3)-generated for q ≠ 4

PSU_3(q^2) are (2,3)-generated for q^2 ≠ 9, 25

PSL_5(q) and PSU_5(q^2) are (2,3)-generated for all q

Abstract

In this paper we determine the classical simple groups of dimension r=3,5 which are (2,3)-generated (the cases r = 2, 4 are known). If r = 3, they are PSL_3(q), q <> 4, and PSU_3(q^2), q^2 <> 9, 25. If r = 5 they are PSL_5(q), for all q, and PSU_5(q^2), q^2 >= 9. Also, the soluble group PSU_3(4) is not (2,3)-generated. We give explicit (2,3)-generators of the linear preimages, in the special linear groups, of the (2,3)-generated simple groups.