Verbal subgroups in the group of triangular matrices over field of characteristic 0.
type: Item
instance of: scholarly article
title: Verbal subgroups in the group of triangular matrices over field of characteristic 0.
zbMATH Open document ID: 1168.20023
author: Agnieszka Bier
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: The author considers the verbal subgroups of the triangular group \(T=Tr(n,F)\) and the unitriangular group \(U=Tr_1(n,F)\), in both cases where \(n\) is a positive integer and \(F\) is a field of characteristic zero. She proves that the only verbal subgroups of \(U\) are the terms of its lower central series, while the verbal subgroups of \(T\) are either the terms of the lower central series of \(U\) or are of the form \(VU\), where \(V\) is a verbal subgroup of the full diagonal group \(D(n,F)\). These results are then extended to the direct limits over \(n\) of the \(Tr(n,F)\) and the \(Tr_1(n,F)\) with the obvious top left corner embeddings. Let \(c_k\) denote the commutator word \([x_1,x_2,\dots,x_k]\) defining the \(k\)-th term of the lower central series of a group. The author's final theorem is that every element of the \(k\)-th term of the lower central series of \(T\), \(U\) and their two direct limits over \(n\) (see above) is a value of \(c_k\) in the group.
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 5545844
zbMATH Keywords: triangular linear groups
zbMATH Keywords: verbal subgroups
zbMATH Keywords: lower central series
reviewed by: B. A. F. Wehrfritz
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.10.006
OpenAlex ID: W2037697147
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DOI: 10.1016/J.JALGEBRA.2008.10.006
description: scientific article; zbMATH DE number 5545844