  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 $q>3$.
