The maximal subgroups of the Monster

Abstract

The classification of the maximal subgroups of the Monster $\mathbf{M}$ is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether $\mathbf{M}$ contains maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(13)$. However, this conclusion relies on reported claims, with unpublished proofs, that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(8)$, $\mathrm{PSL}_2(16)$, or $\mathrm{PSU}_3(4)$. The aim of this paper is to settle all of these questions, and thereby complete the solution to the maximal subgroup problem for $\mathbf{M}$, and for the sporadic simple groups as a whole. Specifically, we prove the existence of two new maximal subgroups of $\mathbf{M}$, isomorphic to the automorphism groups of $\mathrm{PSL}_2(13)$ and $\mathrm{PSU}_3(4)$, and we establish that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSL}_2(8)$ or $\mathrm{PSL}_2(16)$. We also correct the claim that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSU}_3(4)$, and provide evidence that the maximal subgroup $\mathrm{PSL}_2(59)$ (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 $\mathbf{M}$ recently developed by M. Seysen. We provide explicit generators for our newly discovered maximal subgroups of $\mathbf{M}$ in mmgroup format.