A characterisation of almost simple groups with socle ${}^2\E_6(2)$ or $\M(22)$

TL;DR

This paper characterizes the sporadic simple group M(22) and the exceptional group ${}^2E_6(2)$, along with their automorphism groups, using the structure of centralizers of order 3 elements and fusion data.

Contribution

It provides a new characterization of these almost simple groups based on local subgroup structure and fusion information.

Findings

Unique determination of M(22) and ${}^2E_6(2)$ by centralizer structure

Automorphism groups characterized by element fusion data

Advances understanding of group structure from local properties

Abstract

We show that the sporadic simple group $\M(22)$, the exceptional group of Lie type ${}^2\E_6(2)$ and their automorphism groups are uniquely determined by the approximate structure of the centralizer of an element of order 3 together with some information about the fusion of this element in the group.