Intersecting subsets in finite permutation groups
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915079323123712 |
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| author | Li, CaiHeng Pantangi, Venkata Raghu Tej Song, Shujiao Xie, Yilin |
| author_facet | Li, CaiHeng Pantangi, Venkata Raghu Tej Song, Shujiao Xie, Yilin |
| contents | Let $G\leqslant\mathrm{Sym}(Ω)$ be transitive, and let $S$ be an intersecting subset, namely, the ratio $xy^{-1}$ of any elements $x,y\in S$ fixes some point. An EKR-type problem is to characterize transitive groups $G\leqslant\mathrm{Sym}(Ω)$ such that any intersecting set is upper bounded by $|G_ω|$, where $ω\inΩ$. A nice result of Meagher-Spiga-Tiep (2016) tells us that if $G$ is 2-transitive, then indeed $|S|\leqslant|G_ω|$. A natural next step would be to explore intersecting subsets for primitive groups and quasiprimitive groups. Our study in this paper shows that for quasiprimitive permutation groups, the size $|S|$ can be arbitrarily larger than $|G_ω|$. We conjecture that for quasiprimitve groups, the upperbound for $|S|$ is $O(|G_ω||Ω|^{1\over2})$. As a starting point, we prove that ${|S|/(|G_ω||Ω|^{1\over2}})\leqslant{\sqrt2/2}$ for all quasiprimitive actions of the Suzuki groups $G=\mathrm{Sz}(q)$. To show that our conjectured upper bound is tight, we provide examples of groups for which ${|S|/(|G_ω||Ω|^{1\over2}})$ is arbitrarily close to ${\sqrt2/2}$. As far as general transitive groups concerned, infinity families of examples produced show that the ratio ${|S|/(|G_ω||Ω|^{1\over2}})$ can be arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17783 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersecting subsets in finite permutation groups Li, CaiHeng Pantangi, Venkata Raghu Tej Song, Shujiao Xie, Yilin Group Theory Combinatorics 05E18 Let $G\leqslant\mathrm{Sym}(Ω)$ be transitive, and let $S$ be an intersecting subset, namely, the ratio $xy^{-1}$ of any elements $x,y\in S$ fixes some point. An EKR-type problem is to characterize transitive groups $G\leqslant\mathrm{Sym}(Ω)$ such that any intersecting set is upper bounded by $|G_ω|$, where $ω\inΩ$. A nice result of Meagher-Spiga-Tiep (2016) tells us that if $G$ is 2-transitive, then indeed $|S|\leqslant|G_ω|$. A natural next step would be to explore intersecting subsets for primitive groups and quasiprimitive groups. Our study in this paper shows that for quasiprimitive permutation groups, the size $|S|$ can be arbitrarily larger than $|G_ω|$. We conjecture that for quasiprimitve groups, the upperbound for $|S|$ is $O(|G_ω||Ω|^{1\over2})$. As a starting point, we prove that ${|S|/(|G_ω||Ω|^{1\over2}})\leqslant{\sqrt2/2}$ for all quasiprimitive actions of the Suzuki groups $G=\mathrm{Sz}(q)$. To show that our conjectured upper bound is tight, we provide examples of groups for which ${|S|/(|G_ω||Ω|^{1\over2}})$ is arbitrarily close to ${\sqrt2/2}$. As far as general transitive groups concerned, infinity families of examples produced show that the ratio ${|S|/(|G_ω||Ω|^{1\over2}})$ can be arbitrarily large. |
| title | Intersecting subsets in finite permutation groups |
| topic | Group Theory Combinatorics 05E18 |
| url | https://arxiv.org/abs/2403.17783 |