Powers of commutators in linear algebraic groups

Benjamin Martin

doi:10.48550/arXiv.2209.13037

Powers of commutators in linear algebraic groups

Benjamin Martin

Mathematical Science

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

Abstract

Let G be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let G = G(k). We prove that if γ ∈ G such that γ is a commutator and δ ∈ G such that ⟨δ⟩ = ⟨γ⟩ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.

Original language	English
Journal	Proceedings of the Edinburgh Mathematical Society
DOIs	https://doi.org/10.48550/arXiv.2209.13037
Publication status	Accepted/In press - 24 Mar 2024

Bibliographical note

Acknowledgements: I’m grateful to Hendrik Lenstra for introducing me to the Honda property, for stimulating discussions and for sharing a draft of his preprint [9] with me. I’m grateful to Hendrik and to Samuel Tiersma for comments on earlier drafts of this note. I’d also like to thank the referees for some comments and corrections, and the Edinburgh Mathematical Society for supporting Lenstra’s visit to Aberdeen.

Data Availability Statement

Keywords

math.GR
20G15 (20F12, 03C98)

Access to Document

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Cite this

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N2 - Let G be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let G = G(k). We prove that if γ ∈ G such that γ is a commutator and δ ∈ G such that ⟨δ⟩ = ⟨γ⟩ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.

AB - Let G be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a nonarchimedean local field. Let G = G(k). We prove that if γ ∈ G such that γ is a commutator and δ ∈ G such that ⟨δ⟩ = ⟨γ⟩ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz Principle from first-order model theory.

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ER -

Powers of commutators in linear algebraic groups

Abstract

Bibliographical note

Data Availability Statement

Keywords

Access to Document

Embargoed Document

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