Outer automorphism groups of metabelian groups (Q1571065)

Given a group \(G\), the outer automorphism group \(\text{Out }G\) of \(G\) is given as \(\Aut G/\text{Inn }G\), the quotient of the group of all automorphisms by the group of inner automorphisms. There are a number of results about automorphisms of metablian groups which are reviewed here before the proof of the main result. This states that, for any group \(M\), there exists a torsion-free metabelian group \(G\) with trivial centre such that \(M\) is isomorphic to \(\text{Out }G\). The proof involves a long and detailed examination of the situation. In a note added in proof, the authors announce that they have improved the result to: If \(M\) is any group of cardinality \(\kappa<2^{\aleph_0}\) then there is a torsion-free metabelian group \(G\) of cardinality \(|G|=\max\{\kappa,\aleph_0\}\) with \(\text{Out }G\cong M\).