Two examples in the Galois theory of free groups.
type: Item
instance of: scholarly article
title: Two examples in the Galois theory of free groups.
zbMATH Open document ID: 1117.20020
author: Laura Ciobanu
author: Warren Dicks
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: If \(F\) is a free group, \(\Aut(F)\) its group of automorphisms (resp. \(\text{End}(F)\) its monoid of endomorphisms), then there is a correspondence between the subgroups of \(F\) and the subgroups of \(\Aut(F)\) (resp. the submonoids of \(\text{End}(F)\)), in which to every subgroup \(H\) of \(F\) corresponds the subgroup \(\Aut_H(F)\) of \(\Aut(F)\) (resp. the submonoid \(\text{End}_H(F)\) of \(\text{End}(F)\)), which consists of the elements leaving \(H\) elementwise fixed. Also \(\text{Fix}(S)\) for \(S\) a subgroup of \(\Aut(F)\) (resp. submonoid of \(\text{End}(F)\)) denotes the subgroup of \(F\) left elementwise fixed by every element of \(S\). Thus in particular \(\text{Fix}(\Aut_H(F))\) and \(\text{Fix}(\text{End}_H(F))\) are the \(\Aut(F)\)-closure and \(\text{End}(F)\)-closure of \(H\), respectively. \textit{A.~Martino} and \textit{E.~Ventura} [J. Algebra 269, No. 2, 735-747 (2003; Zbl 1035.20018)], have given examples for which \[ \text{Fix}(\Aut_H(F))\neq\text{Fix}(\text{End}_H(F))=H. \] In this present paper, the authors consider two of these examples and for them they give explicit descriptions of \(\text{End}_H(F)\), \(\Aut_H(F)\) and \(\text{Fix}(\Aut_H(F))\) and simplify their proofs. Also, they recall Martino's and Ventura's relevant results and comment on the results contained in the above mentioned paper.
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 5077770
zbMATH Keywords: free groups
zbMATH Keywords: fixed subgroups of automorphisms
zbMATH Keywords: retracts
zbMATH Keywords: fixed subgroups of endomorphisms
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.02.029
OpenAlex ID: W2040847208
cites work: Q4145882
cites work: Intersections of automorphism fixed subgroups in the free group of rank three.
cites work: On automorphism-fixed subgroups of a free group
cites work: Examples of retracts in free groups that are not the fixed subgroup of any automorphism.
cites work: On fixed subgroups of maximal rank
DOI: 10.1016/J.JALGEBRA.2006.02.029
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description: scientific article; zbMATH DE number 5077770