Chirality and non-real elements in $G_2(q)$

TL;DR

This paper investigates the properties of non-real elements in the finite group G_2(q), demonstrating its chirality and contrasting it with the achirality of most classical finite simple groups.

Contribution

It precisely characterizes non-real elements in G_2(q) for certain characteristics and establishes the chirality of this group, a novel insight in group theory.

Findings

G_2(q) is chiral, with a specific word w showing asymmetry.

Most classical finite simple groups are achiral.

Explicit description of non-real elements in G_2(q).

Abstract

In this article, we determine the non-real elements--the ones that are not conjugate to their inverses--in the group $G = G_2(q)$ when $char(F_q)\neq 2,3$. We use this to show that this group is chiral; that is, there is a word w such that $w(G)\neq w(G)^{-1}$. We also show that most classical finite simple groups are achiral