Power maps and completions of free groups and of the modular group
type: Item
instance of: scholarly article
title: Power maps and completions of free groups and of the modular group
zbMATH Open document ID: 0873.20017
author: John G. Thompson
published in: Journal of Algebra
publication date: 1997-11-02T00:00:00Z
review text: Let \(G\) be a group and \(P_n\colon G\to G\), \(x\mapsto x^n\), \(n\in\mathbb{N}\), be the \(n\)-th power map. If \(K\) is a normal subgroup of \(G\) then let \(P_n(G\bmod K)\) be the preimage of \(P_n(G/K)\) in \(G\). If \({\mathcal K}\) is a family of normal subgroups of \(G\) then let be \(P_n(G\bmod{\mathcal K})=\bigcap_{K\in{\mathcal K}}P_n(G\bmod K)\). For \(G\) let \(\text{Fin }G\) be the set of all normal subgroups of \(G\) of finite index. If \(G=F\) is free then \(P_n(F\bmod\text{Fin }F)=P_n(F)\). Now let \(G=\Gamma=\text{SL}(2,\mathbb{Z})\) be the classical modular group. For each \(N\in\mathbb{N}\) let \(\Gamma_N\) be the principal congruence subgroup of level \(N\). If \({\mathcal K}=\{\Gamma_N\mid N\in\mathbb{N}\}\) then \(P_n(\Gamma\bmod{\mathcal K})\subset P_n(\Gamma)\) for \(n>2\) and \(P_n(\Gamma\bmod\text{Fin }\Gamma)=P_n(\Gamma)\) for all \(n\in\mathbb{N}\).
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1019695
zbMATH Keywords: profinite topology
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: power maps
zbMATH Keywords: normal subgroups
zbMATH Keywords: modular group
zbMATH Keywords: principal congruence subgroups
reviewed by: Gerhard Rosenberger
MaRDI profile type: Publication
full work available at URL: jabr.1996.6925
OpenAlex ID: W2093005127
cites work: Q4692374
cites work: Q4145882
DOI: 10.1006/JABR.1996.6925
description: scientific article; zbMATH DE number 1019695