A footnote on residually finite groups
type: Item
instance of: scholarly article
title: A footnote on residually finite groups
zbMATH Open document ID: 0849.20017
DOI: 10.1007/BF02762693
published in: Israel Journal of Mathematics
publication date: 1996-11-05T00:00:00Z
review text: The author gives a short and elementary proof of the following theorem. Let \({\mathfrak S}_n\) be the class of finite soluble groups of rank at most \(n\). Then a finitely generated residually-\(\mathfrak S_n\) group has a nilpotent normal subgroup modulo which the group can be viewed as a matrix group over a direct sum of fields. This result can then be used as a (weak) linearity criterion in place of the much deeper (and usually stronger) criterion of Lubotzky. In particular the theory of finitely generated groups with polynomial subgroup growth, the author claims, can now be developed without recourse to the theory of analytic pro-\(p\) groups.
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 902610
zbMATH Keywords: residually finite groups
zbMATH Keywords: finite soluble groups
zbMATH Keywords: finitely generated residually-\(\mathfrak S_ n\) groups
zbMATH Keywords: nilpotent normal subgroups
zbMATH Keywords: matrix groups over a direct sum of fields
zbMATH Keywords: linearity criterion
zbMATH Keywords: finitely generated groups with polynomial subgroup growth
zbMATH Keywords: analytic pro-\(p\) groups
author: Dan Segal
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 902610