On the orders of primitive groups

Almost all primitive permutation groups of degree n have order at most or have socle isomorphic to a direct power of some alternating group. The Mathieu groups, M11, M12, M23, and M24 are the four exceptions. As a corollary, the sharp version of a theorem of Praeger and Saxl is established, where M12 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.