Using the Mal'cev correspondence for collection in polycyclic groups. (Q2464507)

Let \(G\) be a polycyclic group. It is known that \(G\) has a presentation which is called consistent polycyclic presentation. By this presentation every element of \(G\) has a unique normal form. An algorithm for computing this normal form is called a collection algorithm. Before the appearence of the article under review, the state of the art of computing is the socalled ``collection from the left''. In the article the authors present a method they call ``Mal'cev collection'', using the Mal'cev correspondence which was discovered by A. Mal'cev in 1951. This is in fact a one-to-one correspondence between \(\mathbb{Q}\)-powered nilpotent groups and nilpotent Lie algebras over \(\mathbb{Q}\). Their method is fully implemented in the computer algebra system GAP. In the article, there is a comparison of this method with the existing methods by using different examples for that. A report on the implementation including runtimes for some examples of groups is also given.