International Congress of Mathematicians

Permutation Groups and Normal Subgroups 75Finally we note that Theorem 4.6 is based on the following result about indicesof subgroups of finite simple groups.Theorem 5.5 [5, 30] For a positive real number x, the proportion of integers n < xof the form n = \T : M\, where T is a nonabelian simple group and M is eithera maximal subgroup or a proper subgroup, and (T,M) ^ (A n ,A n^i), is at most(1 + o(lj)c/Ioga:, where c = 1 or c= X^dLi 11(d)res P ec tively.We have presented a framework for studying finite permutation groups byidentifyingand analysing their basic components. The impetus for extending thetheory beyond primitive groups came from the need for an appropriate theory ofbasic permutation groups for combinatorial applications. Developing this theoryrequiredthe answers to specific questions about simple groups, and the power ofthe theory is largely due to its use of the finite simple group classification.References[i[2[3;[4;[6;[r[9[io;[H[12:[is;[w:M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent.Math. 76 (1984), 469^514.R. Baddeley and C. E. Praeger, On primitive overgroups of quasiprimitivepermutation groups, Research Report No. 2002/3, U. Western Australia, 2002.J. Bamberg and C. E. Praeger, Finite permutation groups with a transitiveminimal normal subgroup, preprint, 2002.P. J. Cameron, Finite permutation groups and finite simple groups, Bull. LondonMath. Soc. 13 (1981), 1-22.P. J. Cameron, P. M. Neumann and D. N. Teague, On the degrees of primitivepermutation groups, Math. Z. 180 (1982), 14H49.P. J. Cameron, C. E. Praeger, J. Saxl, and G. M. Seitz, On the Sims conjectureand distance transitive graphs, Bull. London Math. Soc. 15 (1983), 499^506.X. G. Fang, G. Havas, and C. E. Praeger, On the automorphism groups ofquasiprimitive almost simple graphs, J. Algebra 222 (1999), 271^283.X. G. Fang, C. E. Praeger and J. Wang, On the automorphism groups of Cayleygraphs of finite simple groups, J. London Math. Soc. (to appear).M. D. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture,Israel J. Math. 82 (1993), 157^225.M. Giudici, Quasiprimitive groups with no fixed point free elements of primeorder, J. London Math. Soc, (to appear).M. Giudici, C. H. Li and C. E. Praeger, Analysing finite locally «-arc-transitivegraphs, in preparation.A. A. Ivanov, Distance-transitive graphs and their classification, in Investigationsin algebraic theory of combinatorial objects, Kluwer, Dordrecht, 1994,283^378.A. A. Ivanov and C. E. Praeger, On finite affine 2-arc transitive graphs, EuropeanJ. Combin. 14 (1993), 421-144.L. G. Kovacs, Primitive subgroups of wreath products in product action, ProcLondon Math. Soc. (3) 58 (1989), 306^322.