On lattices of closed subgroups in the group of infinite triangular matrices over a field.
type: Item
instance of: scholarly article
title: On lattices of closed subgroups in the group of infinite triangular matrices over a field.
zbMATH Open document ID: 1336.20049
author: Agnieszka Bier
published in: Linear Algebra and its Applications
publication date: 2015-10-13T00:00:00Z
review text: The author investigates a special type of closed subgroups of the topological group \(G=\mathrm{UT}(\infty,K)\) of infinite-dimensional unitriangular matrices over a field \(K\) (\(|K|>2\)), considered with the natural inverse limit topology. Namely, she generalizes the concept of partition subgroups introduced by \textit{A. J. Weir} [Proc. Am. Math. Soc. 6, 454-464 (1955; Zbl 0065.01202)] and defines partition subgroups in \(G\). She shows that they are all closed and discusses the problem of their invariance under various group homomorphisms. She proves that a characteristic subgroup of \(G\) is necessarily a partition subgroup and characterizes the lattices of characteristic and fully characteristic subgroups in \(G\). She concludes with some implications of the given characterization on verbal structure of \(G\) and uses some topological properties to discuss the problem of the width of verbal subgroups in groups defined over a finite field \(K\).
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 6493723
zbMATH Keywords: groups of infinite upper triangular matrices over fields
zbMATH Keywords: infinite unitriangular matrices
zbMATH Keywords: characteristic subgroups
zbMATH Keywords: closed subgroups
zbMATH Keywords: partition subgroups
zbMATH Keywords: widths of verbal subgroups
zbMATH Keywords: lattices of characteristic subgroups
zbMATH Keywords: lattices of fully characteristic subgroups
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: j.laa.2015.07.032
OpenAlex ID: W1117432060
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DOI: 10.1016/J.LAA.2015.07.032
description: scientific article; zbMATH DE number 6493723