Probabilistically nilpotent groups of class two

Sean Eberhard; Pavel  Shumyatsky

doi:10.1007/s00208-023-02567-0

Probabilistically nilpotent groups of class two

Sean Eberhard^*
, Pavel Shumyatsky

^*Corresponding author for this work

School of Mathematics and Physics

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

7 Citations (Scopus)

196 Downloads (Pure)

Abstract

For G a finite group, let d₂(G) denote the proportion of triples (x,y,z) ∈ G³ such that [x,y,z]=1. We determine the structure of finite groups G such that d₂(G) is bounded away from zero: if d₂(G)≥ ϵ >0, G has a class-4 nilpotent normal subgroup H such that [G : H] and |γ₄(H)| are both bounded in terms of ϵ. We also show that if G is an infinite group whose commutators have boundedly many conjugates, or indeed if G satisfies a certain more general commutator covering condition, then G is finite-by-class-3-nilpotent-by-finite.

Original language	English
Pages (from-to)	1879-1902
Number of pages	24
Journal	Mathematische Annalen
Volume	388
Issue number	2
Early online date	25 Jan 2023
DOIs	https://doi.org/10.1007/s00208-023-02567-0
Publication status	Published - Feb 2024

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Probabilistically nilpotent groups of class two
Copyright 2023 The Authors. This is an open access article published under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited.
Final published version, 392 KBLicence: CC BY

Cite this

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N2 - For G a finite group, let d2(G) denote the proportion of triples (x,y,z) ∈ G3 such that [x,y,z]=1. We determine the structure of finite groups G such that d2(G) is bounded away from zero: if d2(G)≥ ϵ >0, G has a class-4 nilpotent normal subgroup H such that [G : H] and |γ4(H)| are both bounded in terms of ϵ. We also show that if G is an infinite group whose commutators have boundedly many conjugates, or indeed if G satisfies a certain more general commutator covering condition, then G is finite-by-class-3-nilpotent-by-finite.

AB - For G a finite group, let d2(G) denote the proportion of triples (x,y,z) ∈ G3 such that [x,y,z]=1. We determine the structure of finite groups G such that d2(G) is bounded away from zero: if d2(G)≥ ϵ >0, G has a class-4 nilpotent normal subgroup H such that [G : H] and |γ4(H)| are both bounded in terms of ϵ. We also show that if G is an infinite group whose commutators have boundedly many conjugates, or indeed if G satisfies a certain more general commutator covering condition, then G is finite-by-class-3-nilpotent-by-finite.

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Probabilistically nilpotent groups of class two

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