Minimal supplements in normalizers of maximal tori of $F_4(q)$

TL;DR

This paper determines the smallest possible size of supplements to maximal tori in the algebraic normalizer of the group F_4(q), identifying cases with complements and minimal lifts of Weyl group elements.

Contribution

It provides the first comprehensive analysis of minimal supplements in normalizers of maximal tori in F_4(q), including explicit minimal orders and lift properties.

Findings

Identified minimal orders of supplements for all maximal tori.

Characterized maximal tori with complements in their normalizers.

Determined minimal lifts of Weyl group elements to the normalizer.

Abstract

Let $G$ be a finite group of Lie type $F_4$ with the Weyl group $W$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement to $T$ in its algebraic normalizer $N(G,T)$. In particular, we obtain all maximal tori having complements in $N(G,T)$. Assume that $T$ corresponds to an element $w$ of $W$. We find the minimal order of lifts of $w$ to $N(G,T)$.