Permutation groups, simple groups, and sieve methods

D. R. Heath-Brown*, Cheryl E. Praeger, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of An-1 in An, 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 Sn and An 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 languageAmerican English
Pages (from-to)347-375
Number of pages29
JournalIsrael Journal of Mathematics
StatePublished - 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


Dive into the research topics of 'Permutation groups, simple groups, and sieve methods'. Together they form a unique fingerprint.

Cite this