Two Theorems on Doubly Transitive Permutation Groups

In a series of papers [3, 4 and 5] on insoluble (transitive) permutation groups of degree p = 2q +1, where p and q are primes, N. Ito has shown that, apart from a small number of exceptions, such a group must be at least quadruply transitive. One of the results which he uses is that an insoluble group of degree p = 2q +1 which is not doubly primitive must be isomorphic to PSL (3, 2) with p = 7. This result is due to H. Wielandt, and ltd gives a proof in [3]. It is quite easy to extend this proof to give the following result: a doubly transitive group of degree 2q + l, where q is prime, which is not doubly primitive, is either sharply doubly transitive or a group of automorphisms of a block design with A = 1 and k = 3. Our notation for the parameters of a block design, v, b, k, r, X, is standard; see [9]. In this paper we shall prove two results about doubly transitive but not doubly primitive groups which resemble the two results mentioned above.