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15 June 2005 On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function
Martin W. Liebeck, Benjamin M. S. Martin, Aner Shalev
Duke Math. J. 128(3): 541-557 (15 June 2005). DOI: 10.1215/S0012-7094-04-12834-9

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'' ζ G 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 , ζ G 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 S n has n o 1 conjugacy classes of primitive maximal subgroups.

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Martin W. Liebeck. Benjamin M. S. Martin. Aner Shalev. "On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function." Duke Math. J. 128 (3) 541 - 557, 15 June 2005. https://doi.org/10.1215/S0012-7094-04-12834-9

Information

Published: 15 June 2005
First available in Project Euclid: 9 June 2005

zbMATH: 1103.20010
MathSciNet: MR2145743
Digital Object Identifier: 10.1215/S0012-7094-04-12834-9

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Rights: Copyright © 2005 Duke University Press

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Vol.128 • No. 3 • 15 June 2005
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