An Erdos-Ko-Rado theorem for the derangement graph of PGL(2,q) acting on the projective line

TL;DR

This paper establishes an Erdős-Ko-Rado type theorem for the derangement graph of PGL(2,q) acting on the projective line, identifying maximum intersecting sets as cosets of point stabilizers.

Contribution

It proves the maximum size of intersecting sets in the derangement graph of PGL(2,q) and characterizes the extremal sets as cosets of stabilizers, extending EKR theory to this group action.

Findings

Maximum size of intersecting sets is q(q-1).

Only cosets of point stabilizers achieve this maximum.

The result generalizes EKR theorems to PGL(2,q) acting on P_q.

Abstract

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 \pi,\sigma in S, there is a projective point p in P_q such that p^\pi=p^\sigma. We prove that if S is intersecting, then the size of S is no more than 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.