Hilbert's 5th problem
type: Item
instance of: scholarly article
title: Hilbert's 5th problem
zbMATH Open document ID: 1335.22008
DOI: 10.4171/LEM/61-1/2-2
author: Lou van den Dries
author: Isaac Goldbring
published in: L'Enseignement Mathématique. 2e Série
publication date: 2016-02-25T00:00:00Z
review text: Assuming a small amount of background, the authors give full proofs of the results by Gleason, Mongomery-Zippin, and Yamabe that characterize Lie groups from among locally compact topological groups. Assuming that \(G\) is a locally compact topological group the following are equivalent: (1) \(G\) is a Lie group. (2) \(G\) has no small subgroups. (3) \(G\) is locally euclidean. (4) \(G\) is locally connected and has no small connected subgroups. (5) \(G\) is locally connected and bounded in dimension. Every compact group \(G\) has an open subgroup that can be approximated by Lie groups, that is, every neighbourhood of the identity of \(G\) contains a compact normal subgroup \(N\) of \(G\) such that \(G/N\) is a Lie group. The paper also gives an account of the work of Hirschfeld, who used nonstandard analysis to simplify some tricky parts of the work by Gleason and Montgomery, and further simplifies them.
reviewed by: Miroslav Repický
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 6545824
zbMATH Keywords: Hilbert's fifth problem
zbMATH Keywords: Lie groups
zbMATH Keywords: no small subgroups
zbMATH Keywords: nonstandard analysis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6545824