Automorphisms of metabelian groups with trivial center (Q1129879)

The authors consider groups which can be realised as the automorphism groups of metabelian groups. In particular they try to answer the question as for which groups \(H\) there exists a metabelian group \(G\) with trivial center such that \(\text{Out }G\cong H\), where \(\text{Out }G\) is the group of outer automorphisms of \(G\). So one of their main results is the following: Let \(B\) be a free metabelian group of rank \(\lambda\) with \(3\leq\lambda<2^{\aleph_0}\). Then there exists a torsion free, complete, metabelian group \(G\) embedding \(B\), with \(G\) containing an abelian characteristic subgroup \(A\) of cardinality \(2^{\aleph_0}\) such that \(G/A\cong B/B'\). We recall that a group is complete if it has trivial center and no outer automorphisms. A group is said to be a unique product group (UP group) if, given any two non-empty finite subsets \(A\) and \(B\) of \(G\), there exists at least one element \(x\) of \(G\) that has a unique representation in the form \(x=ab\) with \(a\in A\) and \(b\in B\). So the authors prove that every abelian group and every UP group can be realized as the outer automorphism group of some metabelian group with trivial center. Their methods include methods from abelian group theory, properties of group rings and the Magnus representation of a free metabelian group as a group of matrices.