An Erdős–Ko–Rado theorem for the derangement graph of acting on the projective line

Let G = PGL ( 2 , q ) be the projective general linear group acting on the projective line P q . A subset S of G is intersecting if for any pair of permutations π , σ in S, there is a projective point p ∈ P q such that p π = p σ . We prove that if S is intersecting, then | S | ⩽ q ( q − 1 ) . Also, we prove that the only sets S that meet this bound are the cosets of the stabilizer of a point of P q .