The binary actions of simple groups of Lie type of characteristic 2
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916760645533696 |
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| author | Gill, Nick Guillot, Pierre Liebeck, Martin W. |
| author_facet | Gill, Nick Guillot, Pierre Liebeck, Martin W. |
| contents | Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $Γ(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For $t\in \mathcal{C}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $Γ(\mathcal{C})$ that lie in the connected component of the graph that contains $t$.
We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The binary actions of simple groups of Lie type of characteristic 2 Gill, Nick Guillot, Pierre Liebeck, Martin W. Group Theory Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $Γ(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For $t\in \mathcal{C}$, we define the component group of $t$ to be the subgroup of $G$ generated by all vertices in $Γ(\mathcal{C})$ that lie in the connected component of the graph that contains $t$. We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group. |
| title | The binary actions of simple groups of Lie type of characteristic 2 |
| topic | Group Theory |
| url | https://arxiv.org/abs/2402.08357 |