Permutation groups,simple groups,and sieve methods |
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Authors: | Email author" target="_blank">D?R?Heath-BrownEmail author Cheryl?E?Praeger Aner?Shalev |
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Institution: | (1) Mathematical Institute, 24-29, St. Giles', OX1 3LB Oxford, England;(2) School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, 6009 Crawley, WA, Australia;(3) Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel |
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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.
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. |
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Keywords: | |
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