Aspects of free groups.
type: Item
instance of: scholarly article
title: Aspects of free groups.
zbMATH Open document ID: 1068.20027
DOI: 10.1016/S0021-8693(02)00665-8
author: André Nies
published in: Journal of Algebra
publication date: 2003-06-09T00:00:00Z
review text: Let \(F_k\) be a free group of rank \(k\). Let \(\overline g\) and \(\overline h\) be \(n\)-tuples of \(F_2\) which satisfy the same existential formulas. It is proved here that there exists an automorphism of \(F_2\) which maps \(\overline g\) to \(\overline h\). The following results are typical: (1) Fix a prime \(p\). \(F_k\) is, up to isomorphism, the only \(k\)-generated group having all finite \(k\)-generated \(p\)-groups as homomorphic images. (2) The theory of non-Abelian free groups has no prime model. Another result worth mentioning is that if the endomorphism \(\alpha\colon F_k\to F_k\) preserves \(\exists\)-formulas in both directions, then \(\alpha\) is an automorphism. -- The reference list contains 12 articles.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03D25
zbMATH DE Number: 1924684
zbMATH Keywords: existential formulas
zbMATH Keywords: free groups
zbMATH Keywords: model theory
zbMATH Keywords: automorphisms
zbMATH Keywords: endomorphisms
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00665-8
OpenAlex ID: W2071077564
description: scientific article; zbMATH DE number 1924684