On the orders of primitive groups

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Almost all primitive permutation groups of degree n have order at most n· ∏ i=0 [ log 2 n]−1 (n−2 i )<n 1+[ log 2 n] , or have socle isomorphic to a direct power of some alternating group. The Mathieu groups, M 11 , M 12 , M 23 , and M 24 are the four exceptions. As a corollary, the sharp version of a theorem of Praeger and Saxl is established, where M 12 turns out to be the “largest” primitive group. For an application, a bound on the orders of permutation groups without large alternating composition factors is given. This sharpens a lemma of Babai, Cameron, Pálfy and generalizes a theorem of Dixon.