Characterization of intersecting families of maximum size in $PSL(2,q)$

TL;DR

This paper characterizes the maximum intersecting families in the group $PSL(2,q)$ acting on a projective line, proving they are all cosets of point stabilizers for odd prime powers greater than 3.

Contribution

It provides a complete characterization of maximum intersecting families in $PSL(2,q)$, showing they are precisely cosets of point stabilizers for all relevant $q$.

Findings

Maximum size of intersecting families is $q(q-1)/2$.

All maximum families are cosets of point stabilizers.

Result holds for all odd prime powers $q > 3$.

Abstract

We consider the action of the $2$-dimensional projective special linear group $PSL(2,q)$ on the projective line $PG(1,q)$ over the finite field $\F_q$, where $q$ is an odd prime power. A subset $S$ of $PSL(2,q)$ is said to be an intersecting family if for any $g_1,g_2 \in S$, there exists an element $x\in PG(1,q)$ such that $x^{g_1}= x^{g_2}$. It is known that the maximum size of an intersecting family in $PSL(2,q)$ is $q(q-1)/2$. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers $q>3$.