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

On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function. (English) Zbl 1103.20010

Duke Math. J. 128, No. 3, 541-557 (2005).
Authors’ summary: 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” \(\zeta_G(s)=\sum_{M\max G}|G:M|^{-s}\), which appears in many probabilistic applications. More specifically, we are able to show that for simple groups \(G\) and for any fixed real number \(s>1\), \(\zeta_G(s)\to 0\) as \(|G|\to\infty\). This confirms a conjecture made by M. W. Liebeck and A. Shalev [in Ann. Math. (2) 144, No. 1, 77-125 (1996; Zbl 0865.20020), p. 84]. We also apply these results to prove the conjecture made by M. W. Liebeck and A. Shalev [in J. Comb. Theory, Ser. A 75, No. 2, 341-352 (1996; Zbl 0866.20003), Conjecture 1, p. 343], that the symmetric group \(S_n\) has \(n^{o(1)}\) conjugacy classes of primitive maximal subgroups.
Reviewer: Victor Mazurov (Novosibirsk)

Cited in 24 Documents

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20E28 Maximal subgroups
11M41 Other Dirichlet series and zeta functions
20E45 Conjugacy classes for groups
20B35 Subgroups of symmetric groups

Keywords:

numbers of conjugacy classes; maximal subgroups of bounded order; finite groups of Lie type; exceptional groups; zeta functions; finite simple groups; symmetric groups

Citations:

Zbl 0865.20020; Zbl 0866.20003
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Full Text: DOI Euclid

References:

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