Automorphism groups of free nilpotent groups
type: Item
instance of: scholarly article
title: Automorphism groups of free nilpotent groups
zbMATH Open document ID: 0638.20023
DOI: 10.1007/BF01194405
author: Chander K. Gupta
author: Roger M. Bryant
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: Let \(F_{n,c}\) be the free nilpotent group of class c and rank n, where c and n are positive integers. \textit{S. Andreadakis} [Arch. Math. 42, 296- 300 (1984; Zbl 0522.20025)] proved that for \(n\geq c\geq 2\) the automorphism group of \(F_{n,c}\) is generated by tame automorphisms (i.e. automorphisms induced by automorphisms of the absolutely free group of rank n) together with certain automorphisms \(\theta _ 3,...,\theta _ c\). In the present paper a simpler result is obtained: for \(n\geq 2\) and \(n\geq c-1\) the automorphism group of \(F_{n,c}\) is generated by tame automorphisms together with the automorphism \(\delta\) satisfying \(x_ 1\delta =x_ 1[x_ 1,x_ 2,x_ 1]\) and \(x_ i\delta =x_ i\) for all \(i>1\), where \(x_ 1,...,x_ n\) are free generators of \(F_{n,c}\). Thus for \(n\geq 4\) and \(n\geq c-1\) the automorphism group of \(F_{n,c}\) is generated by three elements. A more complicated result is also obtained in the case where \(n\geq 2\) and \((c+1)\leq n\leq c-1\).
Mathematics Subject Classification ID: 20F28
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E05
zbMATH DE Number: 4040172
zbMATH Keywords: free nilpotent group
zbMATH Keywords: tame automorphisms
zbMATH Keywords: free generators
reviewed by: Roger M. Bryant
MaRDI profile type: Publication
cites work: On the Automorphisms of Free Groups and Free Nilpotent Groups
cites work: Induced Automorphisms of Free Groups and Free Metabelian Groups
cites work: Die Automorphismengruppe der freien Gruppen
description: scientific article; zbMATH DE number 4040172