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# 1 Introduction

### Sections

1. Setup for computing the correspondence
2. Collection

In this package we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups; it is a correspondence that associates to every Q-powered nilpotent group H a unique rational nilpotent Lie algebra LH and vice-versa. The Mal'cev correspondence was discovered by Anatoly Mal'cev in 1951 Mal51.

## 1.1 Setup for computing the correspondence

Let G be a finitely generated torsion-free nilpotent group, i.e. a T-group. Then G can be embedded in a Q-powered hull hatG. The group hatG is a Q-powered nilpotent group and is unique up to isomorphism. We denote the Lie algebra which corresponds to hatG under the Mal'cev correspondence by L(G)= LhatG.

We provide an algorithm for setting up the Mal'cev correspondence between hatG and the Lie algebra L(G). That is, if G is given by a polycyclic presentation with respect to a Mal'cev basis, then we can compute a structure constants table of L(G). Furthermore for a given ginG we can compute the corresponding element in L(G) and vice versa.

## 1.2 Collection

Every element of a polycyclically presented group has a unique normal form. An algorithm for computing this normal form is called a collection algorithm. Such an algorithm lies at the heart of most methods dealing with polycyclically presented groups. The current state of the art is collection from the left citeGeb02,LGS90,VLe90.

This package contains a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection ALi07. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory.

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