A note on \(IA\)-endomorphisms of two-generated metabelian groups (Q1356823)

Let \(G\) be a group. An endomorphism of \(G\) is called an \(IA\)-endomorphism if it induces the identity map on the factor group \(G/G'\). The set \(IA(G)\) of all \(IA\)-automorphisms of \(G\) is a subgroup of the automorphism group \(\Aut(G)\) of \(G\) that obviously contains the group \(\text{Inn}(G)\) of inner automorphisms of \(G\). Using ring-theoretic methods, the authors give short proofs of some known results about the group of \(IA\)-automorphisms of a \(2\)-generator metabelian group \(G\). In particular, it is proved that in this case \(IA(G)\) is a metabelian group, and that if \(G\) is free metabelian of rank \(2\) then \(IA(G)=\text{Inn}(G)\).