Finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal

TL;DR

This paper classifies certain finite simple exceptional groups of Lie type, specifically E6(q) and 2E6(q), where all subgroups of odd index are pronormal, completing a broader classification effort.

Contribution

It provides a complete classification of finite simple exceptional Lie type groups with all odd index subgroups pronormal, focusing on E6(q) and 2E6(q).

Findings

Classified E6(q) and 2E6(q) groups with all odd index subgroups pronormal.

Completed the classification of all such finite simple exceptional groups of Lie type.

Advances understanding of subgroup structure in these groups.

Abstract

A subgroup $H$ of a group $G$ is said to be pronormal in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. In this paper we classify finite simple groups $E_6(q)$ and ${}^2E_6(q)$ in which all the subgroups of odd index are pronormal. Thus, we complete a classification of finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal.