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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2211.06848 |
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| _version_ | 1866929454170767360 |
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| author | Reid, Colin D. |
| author_facet | Reid, Colin D. |
| contents | Let $Ω$ be a set equipped with an equivalence relation $\sim$; we refer to the equivalence classes as blocks of $Ω$. A permutation group $G \le \mathrm{Sym}(Ω)$ is $k$-by-block-transitive if $\sim$ is $G$-invariant, with at least $k$ blocks, and $G$ is transitive on the set of $k$-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article we classify the finite block-faithful $2$-by-block-transitive actions. We also show that for $k \ge 3$, there are no finite block-faithful $k$-by-block-transitive actions with nontrivial blocks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_06848 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Multiple transitivity except for a system of imprimitivity Reid, Colin D. Group Theory 20B05, 20B20 Let $Ω$ be a set equipped with an equivalence relation $\sim$; we refer to the equivalence classes as blocks of $Ω$. A permutation group $G \le \mathrm{Sym}(Ω)$ is $k$-by-block-transitive if $\sim$ is $G$-invariant, with at least $k$ blocks, and $G$ is transitive on the set of $k$-tuples of points such that no two entries lie in the same block. The action is block-faithful if the action on the set of blocks is faithful. In this article we classify the finite block-faithful $2$-by-block-transitive actions. We also show that for $k \ge 3$, there are no finite block-faithful $k$-by-block-transitive actions with nontrivial blocks. |
| title | Multiple transitivity except for a system of imprimitivity |
| topic | Group Theory 20B05, 20B20 |
| url | https://arxiv.org/abs/2211.06848 |