The (2,3)-generation of the finite 8-dimensional orthogonal groups

TL;DR

This paper proves that certain finite 8-dimensional orthogonal groups can be generated by two elements of orders 2 and 3, with specific conditions on the size of the underlying field.

Contribution

It establishes the (2,3)-generation of the groups _8^+(q) and P_8^+(q) for q , and _8^-(q) and P_8^-(q) for all q , advancing understanding of their algebraic structure.

Findings

_8^+(q) and P_8^+(q) are (2,3)-generated if and only if q .

_8^-(q) and P_8^-(q) are (2,3)-generated for all q .

Abstract

We construct $(2,3)$-generators for the finite $8$-dimensional orthogonal groups, proving the following results: the groups $\Omega_8^+(q)$ and $P\Omega_8^+(q)$ are $(2,3)$-generated if and only if $q\geq 4$; the groups $\Omega_8^-(q)$ and $P\Omega_8^-(q)$ are $(2,3)$-generated for all $q\geq 2$.