On the orders of Primitive Permutation Groups

The problem of bounding the order of a permutation group G in terms of its degree n was one of the central problems of 19th century group theory (see [4]). It is closely related to the 1860 Grand Prix problem of the Paris Academy, but its history goes in fact much further back (see e.g. [3], [1] and [10]). The heart of the problem is of course the case where G is a primitive group. The best result here is due to A. Bochert [2]: if G is primitive and An ^ G then \: G \\ ^ — —!. In the special case where G is not 2-transitive this was greatly improved by Wielandt [13]: in this case \ \\ ^ c " for some constant c (independent of n and G-in fact c = 24 was shown to be sufficient). However, the assumption that G is not 2transitive was essential in Wielandt's approach. It is our aim in this paper to remove this. THEOREM. There is a constant c such that \ \\ < c " for any primitive group G of degree n not containing the alternating group An, and for any n. In fact we can take c = 4. The notation we use is fairly standard (see [12]), but we shall use the symbol GA to denote the setwise and G(A) the pointwise stabilizer in G of a set A. If a and b are (a) integers with b j = 0, then <-> is the least integer m such that m ^ a/b. Also if b is a positive integer and p is a prime, bp will denote the highest integer m for which p m divides b. The proof of the theorem relies on the following two propositions (see [13, Lemma 8.6 and the proof of Theorem 8.5]). PROPOSITION 1. Let G be a primitive group of degree n which is t-transitive but not (t + intransitive, with 1 ^ t < n — 2. Let p be a prime number with p 2 ^ n and let Q be a p-subgroup of G all of whose orbits have length 1 or p. If Q has order p f then PROPOSITION 2. Let G and p be as in Proposition 1. If the Sylow p-subgroups have J,- ft —11 n n order p e then e ^ <> +-= • +