Virtually trivial automorphisms of finitely generated groups (Q1372936)

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 \(\Aut_{vt}(G)\) of all virtually trivial automorphisms of \(G\) is a normal subgroup of the full automorphism group \(\Aut(G)\) of \(G\). Groups for which every automorphism is virtually trivial have been considered by \textit{F. Menegazzo} and \textit{D. J. S. Robinson} [Rend. Semin. Mat. Univ. Padova 78, 267-277 (1987; Zbl 0637.20017)]. Here the authors study the factor group \(\mu(G)=\Aut(G)/\Aut_{vt}(G)\). 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 \(C_{\Aut(A)}(\overline G)\) is finite, where \(\overline G=G/C_G(A)\) is identified with a subgroup of \(\Aut(A)\). Moreover, they provide a method to construct all finitely generated groups \(G\) with \(\mu(G)\) finite.