The (2,3)-generation of the special linear groups over finite fields

TL;DR

This paper classifies which finite special linear groups over finite fields are generated by an involution and an element of order 3, completing the understanding of their (2,3)-generation property.

Contribution

It provides a complete classification of (2,3)-generated finite special linear groups and their projective counterparts, filling a gap in the understanding of their generation properties.

Findings

Identifies all (2,3)-generated $ ext{SL}_n(q)$ groups.

Classifies all (2,3)-generated $ ext{PSL}_n(q)$ groups.

Completes the classification of (2,3)-generation for finite simple groups.

Abstract

We complete the classification of the finite special linear groups $\SL_n(q)$ which are $(2,3)$-generated, i.e., which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups $\PSL_n(q)$ which are $(2,3)$-generated.