On subgroups of finite index in compact Hausdorff groups.
type: Item
instance of: scholarly article
title: On subgroups of finite index in compact Hausdorff groups.
zbMATH Open document ID: 1039.20011
author: D. Massart
published in: Archiv der Mathematik
publication date: 2003-08-18T00:00:00Z
review text: Let \(G\) be a compact Hausdorff group. The authors prove that all abstract subgroups of finite index in \(G\) are open if and only if \(G\) has only finitely many abstract subgroups of index \(n\) for each integer \(n\) if and only if \(G\) has only countably many abstract subgroups of finite index. The proof uses results concerning varieties generated by finite groups and the authors also prove that the statements above are equivalent to the following. Suppose \(V\) is a verbal subgroup of \(G\) corresponding to the variety generated by a finite group then \(V\) is open. The authors remark that some of the interest in results such as these has been generated by J.-P. Serre's well-known conjecture, that in a (topologically) finitely-generated profinite group every subgroup of finite index is open. We note that \textit{N. Nikolav} and \textit{D. Segal} have recently announced a proof of Serre's conjecture [C. R., Math., Acad. Sci. Paris 337, No. 5, 303--308 (2003; Zbl 1033.20029)].
reviewed by: Rachel D. Camina
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1966434
zbMATH Keywords: compact Hausdoff groups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: varieties of groups
zbMATH Keywords: verbal subgroup
zbMATH Keywords: profinite group
MaRDI profile type: Publication
full work available at URL: s00013-003-0786-0
OpenAlex ID: W2080017917
DOI: 10.1007/S00013-003-0786-0
description: scientific article; zbMATH DE number 1966434