IA-automorphisms of finitely generated nilpotent groups. (Q2874691)

An automorphism of a group \(G\) is called an IA-automorphism if it acts trivially on the Abelian factor group \(G/G'\). The set \(\mathrm{IA}(G)\) of all IA-automorphisms of \(G\) is a normal subgroup of the full automorphism group of \(G\), and of course \(\mathrm{IA}(G)\) contains the group \(\mathrm{Inn}(G)\) of all inner automorphisms of \(G\). The subgroup of \(\mathrm{IA}(G)\) consisting of all IA-automorphisms acting trivially on the centre \(Z(G)\) is denote by \(\mathrm{IA}(G)^*\).NEWLINENEWLINE It has recently been proved by \textit{M. S. Attar} [Algebra Colloq. 18, Spec. Iss. 1, 937-944 (2011; Zbl 1297.20020)] that if \(G\) is a finite \(p\)-group of class \(2\), then \(\mathrm{IA}(G)=\mathrm{Inn}(G)\) if and only if \(G'\) is cyclic and \(\mathrm{IA}(G)=\mathrm{IA}(G)^*\).NEWLINENEWLINE In the paper under review, the authors characterize finitely generated nilpotent groups \(G\) of class \(2\) for which \(\mathrm{Inn}(G)\) is isomorphic either to \(\mathrm{IA}(G)\) or to \(\mathrm{IA}(G)^*\). In particular, they prove that if \(G\) is a finitely generated torsion-free nilpotent group of class \(2\), then \(\mathrm{IA}(G)\simeq\mathrm{Inn}(G)\) if and only if \(G'\) is cyclic and the groups \(G/Z(G)\) and \(G/G'\) have the same torsion-free rank (i.e. \(Z(G)/G'\) is finite). Moreover, a finitely generated nilpotent group \(G\) of class \(2\) has cyclic commutator subgroup if and only if \(\mathrm{IA}(G)^*\simeq\mathrm{Inn}(G)\). It follows that if \(G\) is any \(2\)-generator finite nilpotent group of class \(2\), then \(\mathrm{IA}(G)=\mathrm{Inn}(G)\).