The maximal subgroups of the Monster
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916790970351616 |
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| author | Dietrich, Heiko Lee, Melissa Popiel, Tomasz |
| author_facet | Dietrich, Heiko Lee, Melissa Popiel, Tomasz |
| contents | 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 However, this conclusion relies on reported claims, with unpublished proofs, that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(8)$, $\mathrm{PSL}_2(16)$, or $\mathrm{PSU}_3(4)$.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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_14646 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The maximal subgroups of the Monster Dietrich, Heiko Lee, Melissa Popiel, Tomasz Group Theory 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. |
| title | The maximal subgroups of the Monster |
| topic | Group Theory |
| url | https://arxiv.org/abs/2304.14646 |