Groups Acting Generically Multiply Transitively on Solvable Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911847388545024 |
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| author | Berkman, Ayşe Borovik, Alexandre |
| author_facet | Berkman, Ayşe Borovik, Alexandre |
| contents | In this work, we complete the classification of generically multiply transitive actions of groups on solvable groups in the finite Morley rank setting. We prove that if $G$ is a connected group of finite Morley rank acting definably, faithfully and generically $m$-transitively on a connected solvable group $V$ of finite Morley rank where $\operatorname{rk}(V)\leqslant m$, then $\operatorname{rk}(V)=m$, $V$ is a vector space of dimension $m$ over an algebraically closed field $F$, $G\cong \operatorname{GL}_m(F)$, and the action is equivalent to the natural action of $\operatorname{GL}_m(F)$ on $F^m$. This generalises our previous work arXiv:2107.09997. As an application of our result, we classify definably primitive groups of finite Morley rank and affine type acting on a set $X$ with a generic transitivity degree of $\operatorname{rk}(X)+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Groups Acting Generically Multiply Transitively on Solvable Groups Berkman, Ayşe Borovik, Alexandre Group Theory Logic 20F11, 03C60 In this work, we complete the classification of generically multiply transitive actions of groups on solvable groups in the finite Morley rank setting. We prove that if $G$ is a connected group of finite Morley rank acting definably, faithfully and generically $m$-transitively on a connected solvable group $V$ of finite Morley rank where $\operatorname{rk}(V)\leqslant m$, then $\operatorname{rk}(V)=m$, $V$ is a vector space of dimension $m$ over an algebraically closed field $F$, $G\cong \operatorname{GL}_m(F)$, and the action is equivalent to the natural action of $\operatorname{GL}_m(F)$ on $F^m$. This generalises our previous work arXiv:2107.09997. As an application of our result, we classify definably primitive groups of finite Morley rank and affine type acting on a set $X$ with a generic transitivity degree of $\operatorname{rk}(X)+1$. |
| title | Groups Acting Generically Multiply Transitively on Solvable Groups |
| topic | Group Theory Logic 20F11, 03C60 |
| url | https://arxiv.org/abs/2402.06987 |