On splitting of the normalizer of a maximal torus in $E_6(q)$

TL;DR

This paper investigates the structure of maximal tori in the finite group of Lie type E6(q), identifying conditions under which their normalizers split and analyzing the lifts of Weyl group elements.

Contribution

It provides a detailed description of maximal tori with complements in their normalizers in E6(q) and explores the lifting of Weyl group elements when complements do not exist.

Findings

Identifies maximal tori with complements in their normalizers.

Shows that certain Weyl group elements have lifts of the same order.

Provides structural insights into the normalizers of maximal tori in E6(q).

Abstract

Let $G$ be a finite group of Lie type $E_6$ over $F_q$ (adjoint or simply connected) and $W$ be the Weyl group of $G$. We describe maximal tori $T$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. It is well known that for each maximal torus $T$ of $G$ there exists an element $w\in W$ such that $N(G,T)/T\simeq C_W(w)$. When $T$ does not have a complement isomorphic to $C_W(w)$, we show that $w$ has a lift in $N(G,T)$ of the same order.