Elsevier

Journal of Algebra

Volume 258, Issue 2, 15 December 2002, Pages 631-640
Journal of Algebra

On the orders of primitive groups

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Abstract

Almost all primitive permutation groups of degree n have order at most i=0[log2n]−1(n−2i)<n1+[log2n], 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.

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1

On leave from University of Szeged, Hungary. Research partially supported by the Hungarian National Foundation for Scientific Research Grant TO34878.