Transitive permutation groups with trivial four point stabilizers

In this paper we analyze the structure of transitive permutation groups that have trivial four point stabilizers, but some nontrivial three point stabilizer. In particular, we give a complete, detailed classification when the group is simple or quasisimple. This paper is motivated by questions concerning the relationship between fixed points of automorphisms of Riemann surfaces and Weierstra beta points and is a continuation of the authors' earlier work.