Permutation groups, simple groups, and sieve methods

Abstract

We show that the number of integersn≤x which occur as indices of subgroups of nonabelian finite simple groups, excluding that ofA n−1 inA n , is ∼hx/logx, for some given constanth. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indicesn≤x of subgroups of abelian simple groups).We conclude that for most positive integersn, the only quasiprimitive permutation groups of degreen areS n andA n in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982.Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes.