The maximal subgroups of the Monster

Abstract

The classification of the maximal subgroups of the Monster M is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether M contains maximal subgroups that are almost simple with socle ${\mathrm{PSL}}_2\left(13\right)$. However, this conclusion relies on reported claims, with unpublished proofs, that M has no maximal subgroups that are almost simple with socle ${\mathrm{PSL}}_2\left(8\right)$, ${\mathrm{PSL}}_2\left(16\right)$, or ${\mathrm{PSU}}_3\left(4\right)$. The aim of this paper is to settle all of these questions, and thereby complete the solution to the maximal subgroup problem for M, and for the sporadic simple groups as a whole. Specifically, we prove the existence of two new maximal subgroups of M, isomorphic to the automorphism groups of ${\mathrm{PSL}}_2\left(13\right)$ and ${\mathrm{PSU}}_3\left(4\right)$, and we establish that M has no almost simple maximal subgroup with socle ${\mathrm{PSL}}_2\left(8\right)$ or ${\mathrm{PSL}}_2\left(16\right)$. We also correct the claim that M has no almost simple maximal subgroup with socle ${\mathrm{PSU}}_3\left(4\right)$, and provide evidence that the maximal subgroup ${\mathrm{PSL}}_2\left(59\right)$ (constructed in 2004) does not exist. Our proofs are supported by reproducible computations carried out using the publicly available Python package mmgroup for computing with M recently developed by M. Seysen. We provide explicit generators for our newly discovered maximal subgroups of M in mmgroup format.