VOOZH about

URL: https://link.springer.com/article/10.1007/BF02775443?error=cookies_not_supported&code=5cf991db-6e64-4992-94bb-e33ddac8cd86

⇱ Permutation groups, simple groups, and sieve methods | Israel Journal of Mathematics | Springer Nature Link

Skip to main content

Permutation groups, simple groups, and sieve methods

  • Published:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+
from $39.99 /Month
  • Starting from 10 chapters or articles per month
  • Access and download chapters and articles from more than 300k books and 2,500 journals
  • Cancel anytime
View plans

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, books and news in related subjects, suggested using machine learning.

References

  • [BPr] J. Bamberg and C. E. Praeger,Finite permutation groups with a transitive minimal normal subgroup, Proceedings of the London Mathematical Society (3)89 (2004), 71–103.

    Article  MATH  MathSciNet  Google Scholar 

  • [CNT] P. J. Cameron, P. M. Neumann and D. N. Teague,On the degrees of primitive permutation groups, Mathematische Zeitschrift180 (1982), 141–149.

    Article  MATH  MathSciNet  Google Scholar 

  • [C] B. N. Cooperstein,Minimal degree for a permutation representation of a classical group, Israel Journal of Mathematics30 (1978), 213–235.

    Article  MATH  MathSciNet  Google Scholar 

  • [D] H. Davenport,Multiplicative Number Theory, Graduate Texts in Mathematics 74, Springer-Verlag, New York, 2000.

    MATH  Google Scholar 

  • [DM] J. Dixon and B. Mortimer,Permutation Groups, Springer-Verlag, New York, 1996.

    MATH  Google Scholar 

  • [HR] H. Halberstam and H.-E. Richert,Sieve Methods, Academic Press, London, 1974.

    MATH  Google Scholar 

  • [LS1] M. W. Liebeck and J. Saxl,On the orders of maximal subgroups of the finite exceptional groups of Lie type, Proceedings of the London Mathematical Society (3)55 (1987), 299–330.

    Article  MATH  MathSciNet  Google Scholar 

  • [LS2] M. W. Liebeck and J. Saxl,Maximal subgroups of finite simple groups and their automorphism groups, Contemporary Mathematics131 (1992), 243–259.

    MathSciNet  Google Scholar 

  • [NePr] P. M. Neumann and C. E. Praeger,Cyclic matrices over finite fields, Journal of the London Mathematical Society,52 (1995), 263–284.

    MATH  MathSciNet  Google Scholar 

  • [Pr] C. E. Praeger,An O'Nan-Scott Theorem for finite quasiprimitive permutation groups, and an application to 2-arc transitive graphs, Journal of the London Mathematical Society (2)47 (1993), 227–239.

    Article  MATH  MathSciNet  Google Scholar 

  • [PrSh] C. E. Praeger and A. Shalev,Bounds on finite quasiprimitive permutation groups, Journal of the Australian Mathematical Society,71 (2001), 243–258.

    Article  MATH  MathSciNet  Google Scholar 

  • [SS] W. Schwarz and J. Spilker,Arithmetic Functions, London Mathematical Society Lecture Note Series184, Cambridge University Press, Cambridge, 1994.

    MATH  Google Scholar 

  • [T] D. E. Taylor,The Geometry of the Classical Groups, Heldermann Verlag, Berlin, 1992.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, 24-29, St. Giles', OX1 3LB, Oxford, England

    D. R. Heath-Brown

  2. School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, 6009, Crawley, WA, Australia

    Cheryl E. Praeger

  3. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Aner Shalev

Authors
  1. D. R. Heath-Brown
  2. Cheryl E. Praeger
  3. Aner Shalev

Corresponding author

Correspondence to D. R. Heath-Brown.

Additional 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.

About this article

Cite this article

Heath-Brown, D.R., Praeger, C.E. & Shalev, A. Permutation groups, simple groups, and sieve methods. Isr. J. Math. 148, 347–375 (2005). https://doi.org/10.1007/BF02775443

Download citation

  • Received:

  • Revised:

  • Issue date:

  • DOI: https://doi.org/10.1007/BF02775443

Keywords