Permutation Groups with Multiply Transitive Suborbits
Peter J. Cameron · Proceedings of the London Mathematical Society · 1972
In 1929, Manning ([5]) proved that if G is a primitive permutation group on Cl which is not 2-transitive, and if the stabilizer Ga of a point aefiis 2-transitive on an orbit F(a) with | F(a) | = v> 2, then Ga has an orbit A(a) with |A(a) | = w, where w> v and w\\v(v — 1). In [1], I gave a shorter proof of this theorem, using combinatorial techniques developed by Sims, D. G. Higman, and others; I showed that, in the notation of that paper, we can take A = F*°F. Put w = v(v — l)/k, where k is a positive integer. Although Manning's theorem tells us only that k 2. The Mathieu group M22 (or its automorphism group) has a primitive rank 3 representation of degree 77 on the blocks of the associated Steiner system. The subdegrees are 1, 16, 60; the constituent of degree 16 is doubly transitive, and k = 4. The Higman-Sims group HS (or its automorphism group) has a primitive rank 3 representation of degree 100. The subdegrees are 1, 22, 77; the constituent of degree 22 is triply transitive, and k = 6. The purpose of this paper is to obtain a stronger result than that of Manning, and to obtain still stronger results under the assumption that 6?a is more than doubly transitive on F(a). The precise result (which is probably still less than the whole truth) is as follows. Suppose that G is a primitive, not 2-transitive permutation group on Q, and that G ^ is ^-transitive on F(a) (t ^ 2), with |F(a) | = v> 2 and | (F*o F)(a) | = v{v-l)/k. Then the following results hold, (i) jfe<$(«-l); if k = %(v-l) then k = 1 or k = 2. (ii) If t ^ 3, then one of the following is true: (a) k = 1 or k = 2, (b) G has rank 3; |D | = (A+l) 2 (A + 4) 2, v = (A+ 1)(A 2 + 5A + 5), k = (A+l)(A + 2), for some positive integer A. (hi) If t & 4, then k = 1 or k = 2. (iv) If t ^ v — 2 (i.e. if Cra r{a) contains the alternating group) then either (a) k = 1, or (b) k = 2; v is odd; G has degree 2 V ~ 1, rank£(v+ 1)} and has an elementary abelian regular normal subgroup.