The linearity of automorphism groups of relatively free groups
type: Item
instance of: scholarly article
title: The linearity of automorphism groups of relatively free groups
zbMATH Open document ID: 0852.20030
DOI: 10.1007/BF02304781
author: O. M. Mateiko
author: O. I. Tavgen'
published in: Mathematical Notes
publication date: 1996-12-09T00:00:00Z
review text: Let \({\mathcal N}_c\) be the variety of all nilpotent groups of class \(c\) (\(c\geq 1\)), \(\mathcal U\) be the variety of all abelian groups, \({\mathcal U}_k\) be the variety of all abelian groups of exponent \(k\), \({\mathcal B}_m\) be the variety of all locally finite groups of exponent \(m\) and \(\mathcal I\) be the variety of all groups. The following main result is obtained. Theorem. Let \(F_n({\mathcal M})\) be a free group of rank \(n\) of a variety \({\mathcal M}\neq{\mathcal I}\) and \(n\geq 2\). 1. If \({\mathcal M}\nsubseteq{\mathcal N}_c{\mathcal U}{\mathcal B}_m\), then \(\text{Aut }F_n({\mathcal M})\) is not linear. 2. Assume that \({\mathcal M}\subseteq{\mathcal N}_c{\mathcal U}{\mathcal B}_m\) and \({\mathcal U}_k{\mathcal U}\nsubseteq{\mathcal M}\). Then the group \(\text{Aut }F_n({\mathcal M})\) is linear. 3. Assume that \({\mathcal M}\subseteq{\mathcal N}_c{\mathcal U}{\mathcal B}_m\) and \({\mathcal U}_k{\mathcal U}\subseteq{\mathcal M}\). Then \(\text{Aut }F_n({\mathcal M})\) is not linear.
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E10
zbMATH DE Number: 898951
zbMATH Keywords: linear automorphism groups
zbMATH Keywords: relatively free groups
zbMATH Keywords: varieties of groups
zbMATH Keywords: nilpotent groups
zbMATH Keywords: Abelian groups
zbMATH Keywords: locally finite groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 898951