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

We consider the action of the 2-dimensional projective special linear group PSL(2,q) P S L ( 2 , q ) on the projective line PG(1,q) P G ( 1 , q ) over the finite field F q F q , where q is an odd prime power. A subset S of PSL(2,q) P S L ( 2 , q ) is said to be an intersecting family if for any g 1 ,g 2 ∈S g 1 , g 2 ∈ S , there exists an element x∈PG(1,q) x ∈ P G ( 1 , q ) such that x g 1 =x g 2 x g 1 = x g 2 . It is known that the maximum size of an intersecting family in PSL(2,q) P S L ( 2 , q ) is q(q−1)/2 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 q > 3 .