On certain elements of free groups
type: Item
instance of: scholarly article
title: On certain elements of free groups
zbMATH Open document ID: 0912.20021
published in: Journal of Algebra
publication date: 1999-05-25T00:00:00Z
review text: This paper deals with certain elements of a free group \(F_m\), of rank \(m\), on which the way an endomorphism acts determines the endomorphism as a monomorphism or automorphism and the like. More precisely an element \(w\in F_m\) was defined by Shpilrain to be a test word if for any endomorphism \(\varphi\colon F_m\to F_m\) the equality \(\varphi(w)=w\) implies that \(\varphi\) is an automorphism. Several examples of test words are known and are exhibited in this paper. Two definitions are given here generalizing the notion of a test word. Namely: A non-trivial element \(w\in F_m\) is an \(M\)-test word for \(F_m\) if for every endomorphism \(\varphi\) and monomorphism \(\psi\) of \(F_m\) the equation \(\varphi(w)=\psi(w)\) implies that \(\varphi\) is also a monomorphism. A non-trivial word \(w(x_1,\dots,x_n)\) is a \(C\)-test word in \(n\) letters for \(F_m\) if for any two \(n\)-tuples \((A_1,\dots,A_n)\), \((B_1,\dots,B_n)\) of elements of \(F_m\) the equality \(w(A_1,\dots,A_n)=w(B_1,\dots,B_n)\neq 1\) implies the existence of an element \(S\in F_m\) such that \(B_i=SA_iS^{-1}\) for all \(i=1,2,\dots,n\). It follows that any \(C\)-test word in \(n\) letters is both test word and \(M\)-test word for \(F_m\). The purpose of this note is to construct \(C\)-test words in \(n\) letters. So the main result is the following: For an arbitrary \(n\geq 2\) there exists a non-trivial word \(w_n(x_1,\dots,x_n)\) which is a \(C\)-test word in \(n\) letters for any free group \(F_m\) of rank \(m\geq 2\). In addition \(w_n(x_1,\dots,x_n)\) is not a proper power. There are two corollaries of this theorem one of which states that there exist two elements \(u_1,u_2\in F_m\) such that any monomorphism \(\psi\) of \(F_m\) is uniquely determined by \(\psi(u_1)\), \(\psi(u_2)\). The element \(w_n(x_1,\dots,x_n)\) of the theorem is for \(n=2\) \[ \begin{multlined} w_2(x_1,x_2)=[x^8_1,x^8_2]^{100} x_1[x^8_1,x^8_2]^{200} x_1[x^8_1,x^8_2]^{300} x^{-1}_1[x^8_1,x^8_2]x^{-1}_1\\ [x^8_1,x^…
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 1218666
zbMATH Keywords: test words for automorphisms
zbMATH Keywords: test words for monomorphisms
zbMATH Keywords: free groups
zbMATH Keywords: endomorphisms
zbMATH Keywords: \(M\)-test words
zbMATH Keywords: \(C\)-test words
author: S. V. Ivanov
MaRDI profile type: Publication
full work available at URL: jabr.1997.7354
OpenAlex ID: W2039344234
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DOI: 10.1006/JABR.1997.7354
description: scientific article; zbMATH DE number 1218666