Automorphisms of finitely generated free \(G\)-groups (Q1330083)

The author, by generalizing results of \textit{S. Kalajdžievski} [J. Algebra 150, 435-502 (1992; Zbl 0780.20015)] and \textit{S. Krstić} [Proc. Lond. Math. Soc., III. Ser. 64, 49-69 (1992; Zbl 0773.20008)], studies and gives a presentation of the group \(\text{Aut}_ G {\mathcal F}\) of the \(G\)-automorphisms of the \(G\)-free group \(\mathcal F\) of finite rank \(n\). Here \(G\) is a group and the terms \(G\)-group and \(G\)-free group have the usual meaning, i.e. \(\mathcal F\) is a \(G\)-group if there is an action \(x.g\) (with \(x \in {\mathcal F}\), \(g \in G\)) of \(G\) on \(\mathcal F\), and \(\mathcal F\) is \(G\)-free of rank \(n\) if there is a subset \(S\) of \(\mathcal F\) of cardinality \(n\) such that for each \(G\)-group \(\mathcal H\) and each map \(\varphi\) of \(S\) into \(\mathcal H\), there is a unique \(G\)-homomorphism \(\phi\) of \(\mathcal F\) into \(\mathcal H\) with the restriction of \(\phi\) to \(S\) equal to \(\varphi\). A \(G\)- homomorphism \(\phi\) is a homomorphism such that \((x.g)\phi = x \phi.g\) (for \(x \in {\mathcal F}\), \(g\in G\)). The presentation is given taking as generators \(G\)-Whitehead automorphisms of \(F\). These are extensions to \(G\)-free groups of the usual Whitehead automorphisms of the free groups. If \(G\) is an infinite group, the presentation is infinite, but if \(G\) is finitely generated, then \(\text{Aut}_ G{\mathcal F}\) is finitely generated. If \(G\) is finite and \(\mathcal F\) of finite \(G\)-rank \(n\), then the presentation is finite, a result which was proved essentially by Kalajdžievski and Krstić [loc. cit.]. The proof of the main Theorem is based on the Peak Reduction Lemma (PRL) whose proof covers two thirds of the whole paper. The PRL here is an adaption of the usual PRL for free groups used also by the author to give a presentation for the ordinary free group \(F\) [J. Lond. Math. Soc., II. Ser. 8, 259-266 (1974; Zbl 0296.20010)].