Mahammadi Hassanabadi, A.; Robinson, Derek J. S.; Wiegold, James Virtually trivial automorphisms of finitely generated groups. (English) Zbl 0885.20027 Bull. Iran. Math. Soc. 22, No. 1, 35-40 (1996). An automorphism \(\alpha\) of a group \(G\) is called virtually trivial if it acts trivially on a subgroup of finite index of \(G\). The set \(\operatorname{Aut}_{vt}(G)\) of all virtually trivial automorphisms of \(G\) is a normal subgroup of the full automorphism group \(\operatorname{Aut}(G)\) of \(G\). Groups for which every automorphism is virtually trivial have been considered by F. Menegazzo and D. J. S. Robinson [Rend. Semin. Mat. Univ. Padova 78, 267-277 (1987; Zbl 0637.20017)].Here the authors study the factor group \(\mu(G)=\operatorname{Aut}(G)/\operatorname{Aut}_{vt}(G)\). In particular they prove that, if \(G\) is a finitely generated group, the group \(\mu(G)\) is finite if and only if there exists an abelian characteristic subgroup \(A\) of \(G\) with finite index such that \(C_{\operatorname{Aut}(A)}(\overline G)\) is finite, where \(\overline G=G/C_G(A)\) is identified with a subgroup of \(\operatorname{Aut}(A)\). Moreover, they provide a method to construct all finitely generated groups \(G\) with \(\mu(G)\) finite. Reviewer: F.de Giovanni (Napoli) MSC: 20F28 Automorphism groups of groups 20E07 Subgroup theorems; subgroup growth 20F05 Generators, relations, and presentations of groups Keywords:virtually trivial automorphisms; finitely generated groups; subgroups of finite index; automorphism groups; Abelian characteristic subgroups Citations:Zbl 0637.20017 PDFBibTeX XMLCite \textit{A. Mahammadi Hassanabadi} et al., Bull. Iran. Math. Soc. 22, No. 1, 35--40 (1996; Zbl 0885.20027)