On generation by triality automorphisms

TL;DR

This paper investigates the subgroup structures generated by specific automorphisms in certain orthogonal groups, correcting previous work and contributing to the proof of a solvable version of the Baer--Suzuki theorem.

Contribution

It clarifies the structure of subgroups generated by conjugate automorphisms of order 3 in orthogonal groups and corrects prior results impacting the Baer--Suzuki theorem.

Findings

Determined subgroup structures in $O_8^+(2)$ and $O_8^+(3)$

Corrected previous results by S. Guest

Supported the proof of the solvable Baer--Suzuki theorem

Abstract

We clarify the structure of subgroups generated by conjugate graph automorphisms of order $3$ of $O_8^+(2)$ and $O_8^+(3)$. As a result, we obtain a correction to a paper by S. Guest which, in turn, plays an important role in proving the solvable analogue of the Baer--Suzuki theorem.