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

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.