Discrete groups of slow subgroup growth
 Alexander Lubotzky,
 László Pyber,
 Aner Shalev
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 nonpolynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth typen ^{logn } 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 mostn ^{ c logn } subgroups of indexn.
 Jerrum, M. (1986) A compact representation for permutation groups. Journal of Algorithms 7: pp. 3641 CrossRef
 Kantor, M. W. (1979) Permutation representations of the finite classical groups of small degree or rank. Journal of Algebra 60: pp. 158168 CrossRef
 Kleidman, P. B., Liebeck, M. (1990) The Subgroup Structure of the Finite Classical Groups. Cambridge University Press, Cambridge
 Landazuri, V., Seitz, G. M. (1974) On the minimal degrees of projective representations of the finite Chevalley groups. Journal of Algebra 32: pp. 418443 CrossRef
 Liebeck, M. W. (1983) On graphs whose full automorphism group is an alternating group or a finite classical group. Proceedings of the London Mathematical Society 47: pp. 337362 CrossRef
 Lubotzky, A. (1995) Subgroup growth and congruence subgroups. Inventiones mathematicae 119: pp. 267295 CrossRef
 [L2] A. Lubotzky,Subgroup growth, inProc. ICM—Zürich 1994, Birkhäuser Verlag, to appear.
 Lubotzky, A. (1995) Counting finite index subgroups. Groups '93—Galway/St Andrews. Cambridge University Press, Cambridge, pp. 368404
 Lubotzky, A., Mann, A., Segal, D. (1993) Finitely generated groups of polynomial subgroup growth. Israel Journal of Mathematics 82: pp. 363371 CrossRef
 [LPSh] A. Lubotzky, L. Pyber and A. Shalev,Normal and subnormal subgroups in residually finite groups, in preparation.
 Lubotzky, A., Shalev, A. (1994) On some λanalytic prop groups. Israel Journal of Mathematics 85: pp. 307337 CrossRef
 Lubotzky, A., Weiss, B. (1993) Groups and Expanders. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 10: pp. 95109
 Mann, A., Segal, D. (1990) Uniform finiteness conditions in residually finite groups. Proceedings of the London Mathematical Society 61: pp. 529545 CrossRef
 McIver, A., Neumann, P. M. (1987) Enumerating finite groups. The Quarterly Journal of Mathematics, Oxford 38: pp. 473488 CrossRef
 Neumann, B. H. (1937) Some remarks on infinite groups. Journal of the London Mathematical Society 12: pp. 120127 CrossRef
 Praeger, C. E., Saxl, J. (1980) On the orders of primitive permutation groups. The Bulletin of the London Mathematical Society 12: pp. 303307 CrossRef
 Pyber, L. (1993) Asymptotic results for permutation groups. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 11: pp. 197219
 [PSh] L. Pyber and A. Shalev,Groups with superexponential subgroup growth, Combinatorica, to appear.
 Segal, D., Shalev, A. (1993) Groups with fractionally exponential subgroup growth. Journal of Pure and Applied Algebra 88: pp. 205223 CrossRef
 Shalev, A. (1992) Growth functions, padic analytic groups, and groups of finite coclass. Journal of the London Mathematical Society 46: pp. 111122 CrossRef
 [Sh2] A. Shalev,Subgroup growth and sieve methods, preprint.
 Wiegold, J. (1980) Growth sequences of finite groups IV. Journal of the Australian Mathematical Society 29: pp. 1416 CrossRef
 Title
 Discrete groups of slow subgroup growth
 Journal

Israel Journal of Mathematics
Volume 96, Issue 2 , pp 399418
 Cover Date
 19960601
 DOI
 10.1007/BF02937313
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Alexander Lubotzky ^{(1)}
 László Pyber ^{(2)}
 Aner Shalev ^{(3)}
 Author Affiliations

 1. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
 2. Mathematical Institute, Hungarian Academy of Science, P.O.B. 127, H1364, Budapest, Hungary
 3. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel