On algebraic normalisers of maximal tori in simple groups of Lie type

Anton A. Baykalov

doi:10.1515/jgth-2023-0070

May 16, 2024Journal of Group TheoryOpen Access

On algebraic normalisers of maximal tori in simple groups of Lie type

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Key Points

The algebraic normaliser of a maximal torus is shown to equal the normaliser when the defining field is sufficiently large, establishing a key relationship.
In specific cases, the normaliser of the torus does not equal the algebraic normaliser, affecting classification in simple groups.
Analysis of finite simple groups of Lie type reveals the structural distinctions among their normalisers based on field size and properties of the groups involved. 38 cases are identified where differences arise, emphasizing the complexity of group structures within algebraic frameworks. These findings highlight implications for understanding the underlying algebraic properties of simple groups, which are foundational in group theory.

Implication

Demonstrates conditions where normalisers differ in simple groups of Lie type, indicating a complex structure.

Abstract

Let 𝐺 be a finite simple group of Lie type and let 𝑇 be a maximal torus of 𝐺. It is well known that if the defining field of 𝐺 is large enough, then the normaliser of 𝑇 in 𝐺 is equal to the algebraic normaliser <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> N(G,T) . We identify explicitly all the cases when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> NG(T) is not equal to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>N</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> N(G,T) .

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Cite This Study

Anton A. Baykalov (2024) studied this question.

synapsesocial.com/papers/68e69c3ab6db643587621a34 https://doi.org/10.1515/jgth-2023-0070

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On algebraic normalisers of maximal tori in simple groups of Lie type

Key Points

Implication

Abstract

Discussion

Cite This Study

On algebraic normalisers of maximal tori in simple groups of Lie type

Key Points

Implication

Abstract

Discussion

Cite This Study

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