On splitting of the normalizer of a maximal torus in finite groups of Lie type

Abstract

Let $G$ be a finite group of Lie type and $T$ a maximal torus of $G$. In this paper we complete the study of the question of the existence of a complement for the torus $T$ in its algebraic normalizer $N(G,T)$. It is proved that every maximal torus of the group $G\in \{G_2(q),{}^2{G}_2(q),{}^3{D}_4(q)\}$ has a complement in its algebraic normalizer. The remaining twisted classical groups ${}^2{A}_n(q)$ and ${}^2{D}_n(q)$ are also considered.

arXiv comment14 pages