The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers
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arXiv
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| Format: | Preprint |
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2025
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| author | Behajaina, Angelot Maleki, Roghayeh Razafimahatratra, Andriaherimanana Sarobidy |
| author_facet | Behajaina, Angelot Maleki, Roghayeh Razafimahatratra, Andriaherimanana Sarobidy |
| contents | Given a finite transitive group $G\leq \operatorname{Sym}Ω$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of $Ω$, and the order of a point stabilizer of $G$.
In this paper, we completely determine the intersection densities of the permutation groups $\operatorname{PSL}_{2}(q)$, where $q$ is a power of an odd prime $p$, acting transitively with point stabilizers conjugate to $\mathbb{Z}_p$. Our proof uses an auxiliary graph, which is a $\operatorname{PGL}_{2}{q}$-vertex-transitive graph, in which a clique corresponds to an intersecting set of $\operaotnrame{PSL}_{2}(q)$. For the transitive action of $\psl{2}{q}$ with point stabilizers conjugate to $\mathbb{Z}_r$, where $r\mid \frac{q-1}{2}$ is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_00787 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers Behajaina, Angelot Maleki, Roghayeh Razafimahatratra, Andriaherimanana Sarobidy Combinatorics 05C25, 05C69, 05E18, 20B05 Given a finite transitive group $G\leq \operatorname{Sym}Ω$, the {intersection density} of $G$ is defined as the ratio between the size of the largest subsets of $G$ in which any two permutations agree on at least one element of $Ω$, and the order of a point stabilizer of $G$. In this paper, we completely determine the intersection densities of the permutation groups $\operatorname{PSL}_{2}(q)$, where $q$ is a power of an odd prime $p$, acting transitively with point stabilizers conjugate to $\mathbb{Z}_p$. Our proof uses an auxiliary graph, which is a $\operatorname{PGL}_{2}{q}$-vertex-transitive graph, in which a clique corresponds to an intersecting set of $\operaotnrame{PSL}_{2}(q)$. For the transitive action of $\psl{2}{q}$ with point stabilizers conjugate to $\mathbb{Z}_r$, where $r\mid \frac{q-1}{2}$ is an odd prime, we show that the auxiliary graph is not regular, and we construct an intersecting set which is sometimes of maximum size. |
| title | The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers |
| topic | Combinatorics 05C25, 05C69, 05E18, 20B05 |
| url | https://arxiv.org/abs/2511.00787 |