The uniqueness of $PSU_3(8)$ in the Monster

TL;DR

This paper proves that there is a unique conjugacy class of subgroups isomorphic to $PSU_3(8)$ within the Monster group, aiding the classification of its maximal subgroups.

Contribution

It establishes the uniqueness of the conjugacy class of $PSU_3(8)$ subgroups in the Monster, advancing the understanding of its subgroup structure.

Findings

Proves the uniqueness of the $PSU_3(8)$ conjugacy class in the Monster

Uses subgroup computations independent of the Monster itself

Supports the broader goal of classifying maximal subgroups

Abstract

As a contribution to an eventual solution of the problem of determination of the maximal subgroups of the Monster we show that there is a unique conjugacy class of subgroups isomorphic to $PSU_3(8)$. The argument depends on some computations in various subgroups, but not on computations in the Monster itself.