## Abstract

We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of A_{n-1} in A_{n}, is ∼ hx/log x, for some given constant h. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indices n ≤ x of subgroups of abelian simple groups). We conclude that for most positive integers n, the only quasiprimitive permutation groups of degree n are S_{n} and A_{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.

Original language | American English |
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Pages (from-to) | 347-375 |

Number of pages | 29 |

Journal | Israel Journal of Mathematics |

Volume | 148 |

DOIs | |

State | Published - 2005 |

### Bibliographical note

Funding Information:* Research partially supported by the Australian Research Council for C.E.P. and by the Bi-National Science Foundation United States-Israel Grant 2000-053 for A.S. Received January 29, 2004 and in revised form August 1, 2004