The (2,3)-generation of the classical simple groups of dimension 6 and 7

TL;DR

This paper proves that certain finite simple classical groups of dimensions 6 and 7 are generated by two elements of orders 2 and 3, completing their classification for these dimensions.

Contribution

It establishes the (2,3)-generation of specific classical simple groups of dimensions 6 and 7, filling a gap in the classification of such groups.

Findings

Proves $PSp_6(q)$, $ ext{O}_7(q)$, and $PSU_7(q^2)$ are (2,3)-generated for all q.

Completes classification of (2,3)-generated finite classical simple groups up to dimension 7.

Provides new insights into the generation properties of classical groups.

Abstract

In this paper we prove that the finite simple groups $PSp_6(q)$, $\Omega_7(q)$ and $PSU_7(q^2)$ are (2,3)-generated for all q. In particular, this result completes the classification of the (2,3)-generated finite classical simple groups up to dimension 7.