Generators of split extensions of abelian groups by cyclic groups (Q1650832)

Summary: Let \(G\simeq M\rtimes C\) be an \(n\)-generator group which is a split extension of an abelian group \(M\) by a cyclic group \(C\). We study the Nielsen equivalence classes and T-systems of generating \(n\)-tuples of \(G\). The subgroup \(M\) can be turned into a finitely generated faithful module over a suitable quotient \(R\) of the integral group ring of \(C\). When \(C\) is infinite, we show that the Nielsen equivalence classes of the generating \(n\)-tuples of \(G\) correspond bijectively to the orbits of unimodular rows in \(M^{n -1}\) under the action of a subgroup of \(\mathrm{GL}_{n - 1}(R)\). Making no assumption on the cardinality of \(C\), we exhibit a complete invariant of Nielsen equivalence in the case \(M\simeq R\). As an application, we classify Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups, split metacyclic groups and lamplighter groups.