On a conjecture of G.E. Wall

Martin W. Liebeck; Laszlo Pyber; Aner Shalev

doi:10.1016/j.jalgebra.2006.10.047

On a conjecture of G.E. Wall

Martin W. Liebeck, Laszlo Pyber, Aner Shalev^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

26 Scopus citations

Abstract

We prove that if G is a finite almost simple group, having socle of Lie type of rank r, then the number of maximal subgroups of G is at most C r^{- 2 / 3} | G |, where C is an absolute constant. This verifies a conjecture of Wall for groups of sufficiently large rank. Using this we prove that any finite group G has at most 2 C | G |^{3 / 2} maximal subgroups.

Original language	English
Pages (from-to)	184-197
Number of pages	14
Journal	Journal of Algebra
Volume	317
Issue number	1
DOIs	https://doi.org/10.1016/j.jalgebra.2006.10.047
State	Published - 1 Nov 2007

Bibliographical note

Funding Information:
✩ The second author was supported by grants OTKA T049841 and NK62321. The third author acknowledges the support of an EPSRC Visiting Fellowship. * Corresponding author. E-mail address: [email protected] (A. Shalev).

Keywords

Finite groups
Finite simple groups
Maximal subgroups

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10.1016/j.jalgebra.2006.10.047

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abstract = "We prove that if G is a finite almost simple group, having socle of Lie type of rank r, then the number of maximal subgroups of G is at most C r- 2 / 3 | G |, where C is an absolute constant. This verifies a conjecture of Wall for groups of sufficiently large rank. Using this we prove that any finite group G has at most 2 C | G |3 / 2 maximal subgroups.",

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note = "Funding Information: ✩ The second author was supported by grants OTKA T049841 and NK62321. The third author acknowledges the support of an EPSRC Visiting Fellowship. * Corresponding author. E-mail address: [email protected] (A. Shalev).",

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AU - Pyber, Laszlo

AU - Shalev, Aner

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N2 - We prove that if G is a finite almost simple group, having socle of Lie type of rank r, then the number of maximal subgroups of G is at most C r- 2 / 3 | G |, where C is an absolute constant. This verifies a conjecture of Wall for groups of sufficiently large rank. Using this we prove that any finite group G has at most 2 C | G |3 / 2 maximal subgroups.

AB - We prove that if G is a finite almost simple group, having socle of Lie type of rank r, then the number of maximal subgroups of G is at most C r- 2 / 3 | G |, where C is an absolute constant. This verifies a conjecture of Wall for groups of sufficiently large rank. Using this we prove that any finite group G has at most 2 C | G |3 / 2 maximal subgroups.

KW - Finite groups

KW - Finite simple groups

KW - Maximal subgroups

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On a conjecture of G.E. Wall

Abstract

Bibliographical note

Keywords

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