# On splitting of the normalizers of maximal tori in $E_7(q)$ and $E_8(q)$

## TL;DR

This paper classifies maximal tori in groups of Lie type $E_7(q)$ and $E_8(q)$ based on whether they have complements in their normalizers, revealing specific conditions on associated Weyl group elements.

## Contribution

It provides a complete description of when maximal tori in $E_7(q)$ and $E_8(q)$ have complements in their normalizers, including detailed analysis of lifts of Weyl group elements.

## Key findings

- Maximal tori with complements are fully characterized.
- Weyl group elements without lifts of order |w| are identified.
- Exceptions are detailed for the simply-connected $E_7(q)$ group.

## Abstract

Let $G$ be a finite group of Lie type $E_l$ with $l\in\{6,7,8\}$ over $F_q$ and $W$ be the Weyl group of $G$. We describe all maximal tori $T$ of $G$ such that $T$ has a complement in its algebraic normalizer $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. When $T$ does not have a complement, we show that $w$ has a lift in $N(G,T)$ of order $|w|$ in all considered groups, except the simply-connected group $E_7(q)$. In the latter case we describe the elements $w$ that have a lift in $N(G,T)$ of order $|w|$.