Explicit construction of the maximal subgroups of the Monster
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929597699850240 |
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| author | Dietrich, Heiko Lee, Melissa Pisani, Anthony Popiel, Tomasz |
| author_facet | Dietrich, Heiko Lee, Melissa Pisani, Anthony Popiel, Tomasz |
| contents | Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_12230 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit construction of the maximal subgroups of the Monster Dietrich, Heiko Lee, Melissa Pisani, Anthony Popiel, Tomasz Group Theory 20D08, 20E28 Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$. |
| title | Explicit construction of the maximal subgroups of the Monster |
| topic | Group Theory 20D08, 20E28 |
| url | https://arxiv.org/abs/2411.12230 |