Automorphisms in varieties of groups
type: Item
instance of: scholarly article
title: Automorphisms in varieties of groups
zbMATH Open document ID: 0827.20046
published in: Journal of Algebra
publication date: 1995-12-03T00:00:00Z
review text: If \(N\) is a characteristic subgroup of the group \(G\), then each automorphism of \(G\) induces an automorphism on \(G/N\) and so there is a homomorphism \(\pi:\text{Aut}(G)\to\text{Aut}(G/N)\). Thus if \(V\) is a variety of groups, \(V(F_n)\) the verbal subgroup corresponding to \(V\) and \(F_n(V)\cong F_n/V(F)\) the relatively free group of \(V\) of rank \(n\), then there is the homomorphism \(\pi:\text{Aut}(F_n)\to\text{Aut}(F_n(V))\). The automorphisms of \(F_n(V)\) which belong to the image of \(\pi\) are called tame automorphisms of \(F_n(V)\). The problem, which has drawn recently considerable attention, is for which varieties \(V\) and integers \(n\) the group \(F_n(V)\) has non- tame automorphisms. After giving some examples, the authors restrict themselves to considering only IA-automorphisms, i.e. automorphisms which induce the identity automorphism on the derived factor group \(G/G'\). Their main result says that: (i) If \([x_1,x_2,x_1]\) is a law of the variety \(V\), then every IA-automorphism of \(F_n(V)\) is tame and in fact is induced by an IA-automorphism of \(F_n\). (ii) If \([x_1,x_2,x_1]\) is not a law of \(V\) and \(V\) is a variety which is either locally finite or locally nilpotent, then \(F_n(V)\) has non-tame IA- automorphisms for \(n\geq 2\) and one of them is given explicitly in the latter case.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 734565
zbMATH Keywords: automorphisms of relatively free groups
zbMATH Keywords: locally finite variety
zbMATH Keywords: locally nilpotent variety
zbMATH Keywords: variety of groups
zbMATH Keywords: verbal subgroup
zbMATH Keywords: tame automorphisms
zbMATH Keywords: non-tame automorphisms
zbMATH Keywords: IA-automorphisms
zbMATH Keywords: non-tame IA- automorphisms
author: Roger M. Bryant
author: A. I. Papistas
MaRDI profile type: Publication
full work available at URL: jabr.1995.1011
OpenAlex ID: W2056416057
DOI: 10.1006/JABR.1995.1011
description: scientific article; zbMATH DE number 734565