Discrete groups of slow subgroup growth


DOI: 10.1007/BF02937313

Cite this article as:
Lubotzky, A., Pyber, L. & Shalev, A. Israel J. Math. (1996) 96: 399. doi:10.1007/BF02937313


It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least\(n^{\frac{{\log n}}{{\log \log n}}} \). In this paper we prove the existence of a finitely generated group whose subgroup growth is of type\(n^{\frac{{\log n}}{{(\log \log n)^2 }}} \). This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typenlogn is also realized. The proofs involve analysis of the subgroup structure of finite alternating groups and finite simple groups in general. For example, we show there is an absolute constantc such that, ifT is any finite simple group, thenT has at mostnc logn subgroups of indexn.

Copyright information

© Hebrew University 1996

Authors and Affiliations

  • Alexander Lubotzky
    • 1
  • László Pyber
    • 2
  • Aner Shalev
    • 3
  1. 1.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  2. 2.Mathematical InstituteHungarian Academy of ScienceBudapestHungary
  3. 3.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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