Discrete groups of slow subgroup growth

Alexander Lubotzky*, László Pyber, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


It is known that the subgroup growth of finitely generated linear groups is either polynomial or at least {Mathematical expression}. In this paper we prove the existence of a finitely generated group whose subgroup growth is of type {Mathematical expression}. This is the slowest non-polynomial subgroup growth obtained so far for finitely generated groups. The subgroup growth type n log n 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 constant c such that, if T is any finite simple group, then T has at most n c log n subgroups of index n.

Original languageAmerican English
Pages (from-to)399-418
Number of pages20
JournalIsrael Journal of Mathematics
Issue number2
StatePublished - Jun 1996


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