Indeed, the Monster has no almost simple maximal subgroup with socle $\text{PSL}_2(16)$

TL;DR

This paper proves that the Monster group has no maximal subgroups that are almost simple with socle isomorphic to PSL_2(16), supporting a broader classification conjecture about the group's maximal subgroups.

Contribution

The paper provides a proof that the Monster group lacks almost simple maximal subgroups with socle PSL_2(16), using computational methods, advancing the classification of its maximal subgroups.

Findings

No almost simple maximal subgroup with socle PSL_2(16) in the Monster group.

Supports broader conjecture on the classification of Monster's maximal subgroups.

Utilizes computational tools for group theory verification.

Abstract

The classification of the maximal subgroups of the Monster $\mathbf{M}$ is believed to be complete subject to an unpublished result of Holmes and Wilson asserting that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle isomorphic to $\text{PSL}_2(8)$, $\text{PSL}_2(16)$, or $\text{PSU}_3(4)$. We prove this result for $\text{PSL}_2(16)$, with the intention that the other two cases will be dealt with in an expanded version of this paper. Our proof is supported by reproducible computations carried out using Seysen's publicly available Python package mmgroup for computing with $\mathbf{M}$.