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Here we consider  EKR type problems for permutation groups. In particular, we focus on the action  of the 2-dimensional projective special linear group PSL(2,q) on the projective  line PG(1,q) over the finite field ${\mathbb F}_q$, 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  must be cosets of point stabilizers for all odd prime powers q>3. This talk  is based on joint work with Ling Long, Rafael Plaza, and Peter Sin.