Doubly Transitive but Not Doubly Primitive Permutation Groups

The connection between doubly transitive permutation groups G on a finite set Cl which are not doubly primitive and automorphism groups of block designs in which X = 1 has been investigated by Sims [2] and Atkinson [1]. If, for a e Q, Ga has a set of imprimitivity of size 2 then it is easy to show that G is either sharply doubly transitive or is a group of automorphisms of a non-trivial block design on Q in which X = 1. In [1], in the proof of Theorem B, a simple argument due to G. Higman was used to establish the same conclusion if Ga has a set of imprimitivity of size 3. We shall continue the same investigation by proving the following theorem.