On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function

Martin W. Liebeck*, Benjamin M.S. Martin, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

25 Scopus citations

Abstract

We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding "zeta function" ζG(S) = ∑M max G |G : M|-s, which appears in many probabilistic applications. More specifically, we are able to show that for simple groups G and for any fixed real number s > 1, ζG(s) → 0 as |G| → ∞. This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343], that the symmetric group Sn has n°(1) conjugacy classes of primitive maximal subgroups.

Original languageAmerican English
Pages (from-to)541-557
Number of pages17
JournalDuke Mathematical Journal
Volume128
Issue number3
DOIs
StatePublished - 15 Jun 2005

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