Primitive rank 3 groups with a prime subdegree

As a continuation of the study of rank 3 permutation groups G begun in [4] we consider in this paper primitive rank 3 groups of even order in which the stabilizer Ga of a point a has an orbit of prime length. We show in particular that if G has no regular normal subgroup then the minimal normal subgroup M of G is a simple group of rank 3 and the constituent of 214, on the orbit of prime length is nonsolvable and hence doubly transitive. In the first section we present a theorem of WIELANDT on primitive permutation groups (hitherto unpublished) which is important for our discussion and certainly of independent interest. After listing some preliminary facts about rank 3 groups in w 2, we summarize our main results in w 3. The remaining sections contain the proofs of these results, essential use being made in w 4 of a theorem of BRAUER and REYNOLDS [2]. The author is indebted to Professor WIELANDT for communicating the theorem of w 1 and its proof, and for much other valuable help. In particular, the short proof of (3.3) and the method of w 6 are due to Professor WIELANDT. The author is also indebted to Professor J. E. MCLAUGHLIN for many valuable discussions. We take this opportunity to list some corrections to [4] : p. 147 omit the second sentence of Lemma 2. Add to the Cor. to Lemma 3 : Hence { %̀1 for b~F(a) IF(a)~r(b)l= #1 for b~A(a)