  A subset (subgroup) $S$ of a transitive permutation group $G\leq Sym(\Omega)$
is called an intersecting subset (subgroup, resp.) if the ratio $xy^{-1}$ of
any elements $x,y\in S$ fixes some point. A transitive group is said to have
the EKR property if the size of each intersecting subset is at most the order
of the point stabilizer. A nice result of Meagher-Spiga-Tiep (2016) says that
2-transitive permutation groups have the EKR property. In this paper, we
systematically study intersecting subsets in more general transitive
permutation groups, including primitive (quasiprimitive) groups, rank-3 groups,
Suzuki groups, and some special solvable groups. We present new families of
groups that have the EKR property, and various families of groups that do not
have the EKR property. This paper significantly improves the unpublished
version of this paper, and particularly solves Problem 1.4 of it.
