Recognizing automorphisms of the free groups
type: Item
instance of: scholarly article
title: Recognizing automorphisms of the free groups
zbMATH Open document ID: 0802.20024
DOI: 10.1007/BF01196426
author: Vladimir Shpilrain
published in: Archiv der Mathematik
publication date: 1994-12-14T00:00:00Z
review text: If \(F\) is a free group generated freely by the system \(\{x_ i\}_{i \in I}\), where \(I\) is finite or countable infinite, then each mapping \(\varphi\) taking \(x_ i\) to \(y_ i\) of \(F\), for each \(i \in I\), gives an endomorphism of \(F\). This paper is concerned with the problem of deciding whether or not this endomorphism \(\varphi\) is actually an automorphism. There are the classical algorithms by Nielsen and Whitehead and also the more recent ``inverse function theorem'' by \textit{J. S. Birman} [Proc. Am. Math. Soc. 41, 634-638 (1973; Zbl 0274.20032)], which are applied in the case where the \(y_ i\) are given explicitly. In the case of two generators there is Nielsen's ``commutator test'' which says that \(\varphi\) is an automorphism if \([y_ 1,y_ 2]\) is conjugate to \([x_ 1,x_ 2]\) or its inverse. The main result of the present paper gives a generalization of Nielsen's test. More precisely it is proved: Let \(R\) be an arbitrary subgroup of a free group \(F\). Then: (a) generators \(x_ 1,\dots,x_ n\) of \(F\) belong to \(R\) if and only if the element \(u_ 1(n)=[x_ 1,\dots,x_ n]\) belongs to \(\gamma_ n(R)\); (b) for \(n=2m\), generators \(x_ 1,\dots,x_ n\) of \(F\) belong to \(R\) if and only if the element \(u_ 2(2m)=[x_ 1,x_ 2]\dots [x_{2m- 1},x_{2m}]\) belongs to \(R^ 2\); (c) for an arbitrary \(k \geq 2\), generators \(x_ 1,\dots,x_ n\) of \(F\) belong to \(R\) if and only if the element \(u_ 3(n,k)=x^ k_ 1 \dots x^ k_ x\) belongs to \(R' R^ k\). One of the corollaries states that if \(\varphi\) is an endomorphism of \(F=F_ n\), \(n \geq 2\) and \(u\) is one of the words \(u_ 1(n)\), \(u_ 2(2m)\), \(u_ 3(n,k)\) then \(\varphi\) is an automorphism if and only if \(\varphi(u)=\alpha(u)\) for some \(\alpha \in \text{Aut }F_ n\). It is also proved that if \(N\) is a fully invariant subgroup of the free group \(F=F_ n\), \(n=2m\), \(G=F/[N',F]\) and \(\varphi\) an endomorphism of \(G\) such that \(\varphi(u_ 2(2m))=\alpha(u_ 2(2m))\) for some \(\alp…
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 590065
zbMATH Keywords: inverse function theorem
zbMATH Keywords: commutator test
zbMATH Keywords: free groups
zbMATH Keywords: endomorphisms
zbMATH Keywords: automorphisms
zbMATH Keywords: generators
zbMATH Keywords: fully invariant subgroups
zbMATH Keywords: group rings
zbMATH Keywords: free derivatives
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01196426
OpenAlex ID: W2000886002
description: scientific article; zbMATH DE number 590065