Irredundant bases for the symmetric group
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910375487733760 |
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| author | Roney-Dougal, Colva M. Wu, Peiran |
| author_facet | Roney-Dougal, Colva M. Wu, Peiran |
| contents | An irredundant base of a group $G$ acting faithfully on a finite set $Γ$ is a sequence of points in $Γ$ that produces a strictly descending chain of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup. Suppose that $G$ is $\operatorname{S}_n$ or $\operatorname{A}_n$ acting primitively on $Γ$, and that the point stabiliser is primitive in its natural action on $n$ points. We prove that the maximum size of an irredundant base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log n)^2\right)$. We also show that these bounds are best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_00092 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Irredundant bases for the symmetric group Roney-Dougal, Colva M. Wu, Peiran Group Theory 20B15 (Primary) 20D06, 20E15 (Secondary) An irredundant base of a group $G$ acting faithfully on a finite set $Γ$ is a sequence of points in $Γ$ that produces a strictly descending chain of pointwise stabiliser subgroups in $G$, terminating at the trivial subgroup. Suppose that $G$ is $\operatorname{S}_n$ or $\operatorname{A}_n$ acting primitively on $Γ$, and that the point stabiliser is primitive in its natural action on $n$ points. We prove that the maximum size of an irredundant base of $G$ is $O\left(\sqrt{n}\right)$, and in most cases $O\left((\log n)^2\right)$. We also show that these bounds are best possible. |
| title | Irredundant bases for the symmetric group |
| topic | Group Theory 20B15 (Primary) 20D06, 20E15 (Secondary) |
| url | https://arxiv.org/abs/2309.00092 |