Normalisers of maximal tori and a conjecture of Vdovin

Tim  Burness; Adam Thomas

doi:10.1016/j.jalgebra.2022.12.013

Normalisers of maximal tori and a conjecture of Vdovin

Tim Burness^*
, Adam Thomas^*

^*Corresponding author for this work

Research output: Contribution to journal › Article (Academic Journal) › peer-review

Abstract

Abstract. Let G = O p 0 (G¯F ) be a finite simple group of Lie type defined over a field of characteristic p, where F is a Steinberg endomorphism of the ambient simple algebraic group G¯. Let T¯ be an F-stable maximal torus of G¯ and set N = NG(T¯). A conjecture due to Vdovin asserts that if G 6∼= L3(2) then N ∩ N x is a p-group for some x ∈ G. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of G on G/N, which allows us to prove a stronger, and suitably modified, version of Vdovin’s conjecture.

Original language	English
Pages (from-to)	459-504
Number of pages	46
Journal	Journal of Algebra
Volume	619
Early online date	5 Jan 2023
DOIs	https://doi.org/10.1016/j.jalgebra.2022.12.013
Publication status	Published - 1 Apr 2023

Bibliographical note

Funding Information:
Acknowledgements. Burness thanks the Department of Mathematics at the University of Padua for their generous hospitality during a research visit in autumn 2021. Thomas is supported by EPSRC grant EP/W000466/1 .

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© 2022 The Author(s)

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title = "Normalisers of maximal tori and a conjecture of Vdovin",

abstract = "Abstract. Let G = O p 0 (G¯F ) be a finite simple group of Lie type defined over a field of characteristic p, where F is a Steinberg endomorphism of the ambient simple algebraic group G¯. Let T¯ be an F-stable maximal torus of G¯ and set N = NG(T¯). A conjecture due to Vdovin asserts that if G 6∼= L3(2) then N ∩ N x is a p-group for some x ∈ G. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of G on G/N, which allows us to prove a stronger, and suitably modified, version of Vdovin{\textquoteright}s conjecture.",

author = "Tim Burness and Adam Thomas",

note = "Funding Information: Acknowledgements. Burness thanks the Department of Mathematics at the University of Padua for their generous hospitality during a research visit in autumn 2021. Thomas is supported by EPSRC grant EP/W000466/1 . Publisher Copyright: {\textcopyright} 2022 The Author(s)",

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AU - Burness, Tim

AU - Thomas, Adam

N1 - Funding Information: Acknowledgements. Burness thanks the Department of Mathematics at the University of Padua for their generous hospitality during a research visit in autumn 2021. Thomas is supported by EPSRC grant EP/W000466/1 . Publisher Copyright: © 2022 The Author(s)

PY - 2023/4/1

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N2 - Abstract. Let G = O p 0 (G¯F ) be a finite simple group of Lie type defined over a field of characteristic p, where F is a Steinberg endomorphism of the ambient simple algebraic group G¯. Let T¯ be an F-stable maximal torus of G¯ and set N = NG(T¯). A conjecture due to Vdovin asserts that if G 6∼= L3(2) then N ∩ N x is a p-group for some x ∈ G. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of G on G/N, which allows us to prove a stronger, and suitably modified, version of Vdovin’s conjecture.

AB - Abstract. Let G = O p 0 (G¯F ) be a finite simple group of Lie type defined over a field of characteristic p, where F is a Steinberg endomorphism of the ambient simple algebraic group G¯. Let T¯ be an F-stable maximal torus of G¯ and set N = NG(T¯). A conjecture due to Vdovin asserts that if G 6∼= L3(2) then N ∩ N x is a p-group for some x ∈ G. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of G on G/N, which allows us to prove a stronger, and suitably modified, version of Vdovin’s conjecture.

U2 - 10.1016/j.jalgebra.2022.12.013

DO - 10.1016/j.jalgebra.2022.12.013

M3 - Article (Academic Journal)

SN - 0021-8693

VL - 619

SP - 459

EP - 504

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Normalisers of maximal tori and a conjecture of Vdovin

Abstract

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