Asymptotics of the powers in finite reductive groups
type: Item
instance of: scholarly article
title: Asymptotics of the powers in finite reductive groups
author: Amit Kulshrestha
author: Rijubrata Kundu
author: Anupam Singh
published in: Journal of Group Theory
publication date: 2022-11-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/2004.12616
review text: The word maps on finite groups of Lie type and algebraic groups have been studied extensively in the last years. Let \(G\) be a connected reductive group over an algebraic closure of a finite field \(\mathbb{F}_q\). Let \(F\) be a Frobenius map on \(G\) giving rise to a finite group of Lie type with \(\mathbb{F}_q\)-points given by \(G(\mathbb{F}_q)=G^F\). Let \(M\geq 2\) be a positive integer. Consider the power map \(G\to G\) given by \(x\to x^M\). In this paper, the authors study asymptotic estimates of the probability that a regular semisimple, semisimple and regular element is an \(M\)th power in \(G(\mathbb{F}_q)\). They prove that, as \(q\to\infty\), the set of limits for each proportion of \(M\)th powers of regular semisimple, semisimple and regular elements of \(G(\mathbb{F}_q)\) is the same and provide a formula. This generalizes the well-known results for \(M=1\) where all the limits take the same value 1. The authors also compute this more explicitly for the groups \(\mathrm{GL}(n,q)\) and \(U(n,q)\) and show that the set of limits is the same for these two group, in fact, in bijection under \(q\mapsto -q\) for a fixed \(M\).
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 20P05
zbMATH DE Number: 7612818
zbMATH Keywords: finite reductive group
zbMATH Keywords: power map
zbMATH Keywords: asymptotics of the powers
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
OpenAlex ID: W3199565012
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DOI: 10.1515/JGTH-2020-0206
description: scientific article; zbMATH DE number 7612818