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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.09636 |
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| _version_ | 1866916567743201280 |
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| author | Altınel, Tuna Wiscons, Joshua |
| author_facet | Altınel, Tuna Wiscons, Joshua |
| contents | We show that a generically sharply $t$-transitive permutation group of finite Morley rank on a set of rank $r$ satisfies $t\le r+2$ provided the pointwise stabilizer of a generic $(t-1)$-tuple is an $L$-group, which holds, for example, when this stabilizer is solvable or when $r\le 5$. This makes progress on the Borovik-Cherlin conjecture that every generically $(r+2)$-transitive permutation group of finite Morley rank on a set of rank $r$ is of the form $\operatorname{PGL}_{r+1}(F)$ acting naturally on $\mathbb{P}^r(F)$.
Our proof is assembled from three key ingredients that are independent of the main theorem - these address actions of $\operatorname{Alt}(n)$ on $L$-groups of finite Morley rank, generically $2$-transitive actions with abelian point stabilizers, and simple groups of rank $6$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounding the degree of generic sharp transitivity Altınel, Tuna Wiscons, Joshua Group Theory Logic 20B22 (Primary) 20F11 (Secondary) We show that a generically sharply $t$-transitive permutation group of finite Morley rank on a set of rank $r$ satisfies $t\le r+2$ provided the pointwise stabilizer of a generic $(t-1)$-tuple is an $L$-group, which holds, for example, when this stabilizer is solvable or when $r\le 5$. This makes progress on the Borovik-Cherlin conjecture that every generically $(r+2)$-transitive permutation group of finite Morley rank on a set of rank $r$ is of the form $\operatorname{PGL}_{r+1}(F)$ acting naturally on $\mathbb{P}^r(F)$. Our proof is assembled from three key ingredients that are independent of the main theorem - these address actions of $\operatorname{Alt}(n)$ on $L$-groups of finite Morley rank, generically $2$-transitive actions with abelian point stabilizers, and simple groups of rank $6$. |
| title | Bounding the degree of generic sharp transitivity |
| topic | Group Theory Logic 20B22 (Primary) 20F11 (Secondary) |
| url | https://arxiv.org/abs/2407.09636 |