Permutation groups, simple groups, and sieve methods

D. R. Heath-Brown; Cheryl E. Praeger; Aner Shalev

doi:10.1007/BF02775443

Permutation groups, simple groups, and sieve methods

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

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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	English
Pages (from-to)	347-375
Number of pages	29
Journal	Israel Journal of Mathematics
Volume	148
DOIs	https://doi.org/10.1007/BF02775443
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

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title = "Permutation groups, simple groups, and sieve methods",

abstract = "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.",

author = "Heath-Brown, \{D. R.\} and Praeger, \{Cheryl E.\} and Aner Shalev",

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",

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doi = "10.1007/BF02775443",

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T1 - Permutation groups, simple groups, and sieve methods

AU - Heath-Brown, D. R.

AU - Praeger, Cheryl E.

AU - Shalev, Aner

N1 - 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

PY - 2005

Y1 - 2005

N2 - 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.

AB - 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.

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Permutation groups, simple groups, and sieve methods

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