Conjugacy classes of derangements in finite groups of Lie type

Sean Eberhard; Daniele Garzoni

doi:10.1090/btran/193

Conjugacy classes of derangements in finite groups of Lie type

Sean Eberhard, Daniele Garzoni

School of Mathematics and Physics

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Abstract

Let G be a finite almost simple group of Lie type acting faithfully and primitively on a set Ω. We prove an analogue of the Boston–Shalevconjecture for conjugacy classes: the proportion of conjugacy classes of G consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).

Original language	English
Pages (from-to)	536-575
Number of pages	40
Journal	Transactions of the American Mathematical Society: Series B
Volume	12
DOIs	https://doi.org/10.1090/btran/193
Publication status	Published - 22 Apr 2025

Keywords

conjugacy classes
derangements
finite groups

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10.1090/btran/193Licence: CC BY

Conjugacy classes of derangements in finite groups of Lie type
Copyright 2025 The Authors. This is an open access article published under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/3.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited.
Final published version, 468 KBLicence: CC BY

Cite this

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abstract = "Let G be a finite almost simple group of Lie type acting faithfully and primitively on a set Ω. We prove an analogue of the Boston–Shalevconjecture for conjugacy classes: the proportion of conjugacy classes of G consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry). ",

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N2 - Let G be a finite almost simple group of Lie type acting faithfully and primitively on a set Ω. We prove an analogue of the Boston–Shalevconjecture for conjugacy classes: the proportion of conjugacy classes of G consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).

AB - Let G be a finite almost simple group of Lie type acting faithfully and primitively on a set Ω. We prove an analogue of the Boston–Shalevconjecture for conjugacy classes: the proportion of conjugacy classes of G consisting of derangements is bounded away from zero. This answers a question of Guralnick and Zalesski. The proof is based on results on the anatomy of palindromic polynomials over finite fields (with either reflective symmetry or conjugate-reflective symmetry).

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KW - derangements

KW - finite groups

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JO - Transactions of the American Mathematical Society: Series B

JF - Transactions of the American Mathematical Society: Series B

ER -

Conjugacy classes of derangements in finite groups of Lie type

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Keywords

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