Stability for Intersecting Families in PGL(2, q)

We consider the action of the 2-dimensional projective general linear group PGL(2, q) on the projective line PG(1, q). A subset S of PGL(2, q) is said to be an intersecting family if for every g1, g2 ∈ S, there exists α ∈ PG(1, q) such that αg1 = αg2 . It was proved by Meagher and Spiga that the intersecting families of maximum size in PGL(2, q) are precisely the cosets of point stabilizers. We prove that if an intersecting family S ⊂ PGL(2, q) has size close to the maximum then it must be “close” in structure to a coset of a point stabilizer. This phenomenon is known as stability. We use this stability result proved here to show that if the size of S is close enough to the maximum then S must be contained in a coset of a point stabilizer.