Given a finite transitive group \(G\le \operatorname {Sym}(\Omega )\) , 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 \(\operatorname {PSL}_2(q)\) . For the transitive action of \(\operatorname {PSL}_2(q)\) 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.
