On endomorphisms of free groups that preserve primitivity
type: Item
instance of: scholarly article
title: On endomorphisms of free groups that preserve primitivity
zbMATH Open document ID: 0926.20018
published in: Archiv der Mathematik
publication date: 1999-12-01T00:00:00Z
review text: An element of a free group \(F_n\) of rank \(n\) is called primitive if it can be included in a free basis of \(F_n\). This paper deals with endomorphisms of free groups of finite rank which preserve primitivity. It is proved that in many cases these endomorphisms are automorphisms. The main results include the theorem that if \(\varphi\) is an endomorphism of the free group \(F_2\), of rank 2, which preserves primitivity, i.e. maps a primitive element onto a primitive element, then \(\varphi\) is an automorphism. If the image \(\varphi(F_n)\) of an endomorphism \(\varphi\) which preserves primitivity contains a primitive pair (i.e. two independent elements which can be included in a basis of \(F_n\)), then the above theorem holds for arbitrary finite ranks \(n\geq 2\). The main theorems are stated and proved in a more general situation. The proofs are carried out combinatorially using cancellation arguments and the Whitehead automorphisms.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 1295479
zbMATH Keywords: free bases
zbMATH Keywords: endomorphisms of free groups of finite rank
zbMATH Keywords: primitive elements
zbMATH Keywords: Whitehead automorphisms
author: S. V. Ivanov
MaRDI profile type: Publication
full work available at URL: s000130050309
OpenAlex ID: W2007789893
DOI: 10.1007/S000130050309
description: scientific article; zbMATH DE number 1295479