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

TL;DR

This paper investigates when the normalizer of a maximal torus in finite groups of Lie type splits, providing complete results for certain groups and partial results for others.

Contribution

It completes the classification of splitting of maximal tori in specific finite groups of Lie type, including new results for twisted classical groups.

Findings

Maximal tori in G2(q), 2G2(q), and 3D4(q) have complements in their normalizers.

Partial results are obtained for twisted classical groups 2A_n(q) and 2D_n(q).

The study advances understanding of the structure of algebraic normalizers 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 $G\in\{G_2(q), {}^2G_2(q), {}^3D_4(q)\}$ has a complement in its algebraic normalizer. The remaining twisted classical groups ${}^2A_n(q)$ and ${}^2D_n(q)$ are also considered.