Chirality and 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 $char(F_q)\ne 2,3$. We use this to show that this group is chiral; that is, there is a word w such that $w(G)\ne w(G{)}^{-1}$. We also show that most classical finite simple groups are achiral

13 pages. Keywords: Chirality, word maps, non-real elements, an exceptional group of Lie type G_2(q)