Two examples in the Galois theory of free groups.
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.
scientific article; zbMATH DE number 5077770