Some Properties of Twisted Chevalley Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913867433508864 |
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| author | Makadiya, Deep H. |
| author_facet | Makadiya, Deep H. |
| contents | This thesis investigates certain structural properties of twisted Chevalley groups over commutative rings, focusing on three key problems.
Let $R$ be a commutative ring satisfying mild conditions. Let $G_{π,σ} (Φ, R)$ denote a twisted Chevalley group over $R$, and let $E'_{π, σ} (Φ, R)$ denote its elementary subgroup. The first problem concerns the normality of $E'_{π, σ} (Φ, R, J)$, the relative elementary subgroups at level $J$, in the group $G_{π, σ} (Φ, R)$. The second problem addresses the classification of the subgroups of $G_{π, σ}(Φ, R)$ that are normalized by $E'_{π, σ}(Φ, R)$. This classification provides a comprehensive characterization of the normal subgroups of $E'_{π, σ}(Φ, R)$. Lastly, the third problem investigates the normalizers of $E'_{π, σ}(Φ, R)$ and $G_{π, σ}(Φ, R)$ in the bigger group $G_{π, σ}(Φ, S)$, where $S$ is a ring extension of $R$. We prove that these normalizers coincide. Moreover, for groups of adjoint type, we show that they are precisely equal to $G_{π, σ}(Φ, R)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24430 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some Properties of Twisted Chevalley Groups Makadiya, Deep H. Group Theory 20G35, 17B20, 17B22 This thesis investigates certain structural properties of twisted Chevalley groups over commutative rings, focusing on three key problems. Let $R$ be a commutative ring satisfying mild conditions. Let $G_{π,σ} (Φ, R)$ denote a twisted Chevalley group over $R$, and let $E'_{π, σ} (Φ, R)$ denote its elementary subgroup. The first problem concerns the normality of $E'_{π, σ} (Φ, R, J)$, the relative elementary subgroups at level $J$, in the group $G_{π, σ} (Φ, R)$. The second problem addresses the classification of the subgroups of $G_{π, σ}(Φ, R)$ that are normalized by $E'_{π, σ}(Φ, R)$. This classification provides a comprehensive characterization of the normal subgroups of $E'_{π, σ}(Φ, R)$. Lastly, the third problem investigates the normalizers of $E'_{π, σ}(Φ, R)$ and $G_{π, σ}(Φ, R)$ in the bigger group $G_{π, σ}(Φ, S)$, where $S$ is a ring extension of $R$. We prove that these normalizers coincide. Moreover, for groups of adjoint type, we show that they are precisely equal to $G_{π, σ}(Φ, R)$. |
| title | Some Properties of Twisted Chevalley Groups |
| topic | Group Theory 20G35, 17B20, 17B22 |
| url | https://arxiv.org/abs/2505.24430 |