The intersection densities of transitive actions of $\operatorname{PSL}_{2}(q)$ with cyclic point stabilizers

TL;DR

This paper determines the intersection densities of certain permutation groups $ ext{PSL}_2(q)$ with cyclic point stabilizers, using graph-theoretic methods to analyze intersecting sets and their maximal sizes.

Contribution

It provides a complete characterization of intersection densities for $ ext{PSL}_2(q)$ acting transitively with specific cyclic point stabilizers, employing an auxiliary graph approach.

Findings

Intersection densities are explicitly determined for $ ext{PSL}_2(q)$ with point stabilizers conjugate to $ ext{Z}_p$.

The auxiliary graph used is a $ ext{PGL}_2(q)$-vertex-transitive graph, linking cliques to intersecting sets.

For certain stabilizers, the auxiliary graph is not regular, and maximum intersecting sets are constructed.

Abstract

Given a finite transitive group $G\leq \operatorname{Sym}{\Omega}$, 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 $\Omega$, 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.