Algorithmic use of the Mal'cev correspondence

Summary

Abstract

Mal'cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically presented groups.

Introduction

The connection between groups and Lie rings, respectively Lie algebras, is a wellknown and mathematically very useful concept. For example, a typical way to solve a problem in a Lie group is to transfer the problem to the Lie algebra of the group, study it there with the help of tools from linear algebra and transfer the result back into the Lie group.

Mechanisms of this kind have also been shown to be useful for algorithmic applications. For instance, Vaughan-Lee and O'Brien used Lie ring techniques to construct a consistent polycyclic presentation of R (2, 7), the largest 2-generator group of exponent 7 [16].

In this paper we demonstrate the algorithmic usefulness of the Mal'cev correspondence between torsion-free radicable nilpotent groups and rational nilpotent Lie algebras. We use it to develop an algorithm for fast multiplication in polycyclic groups. To be more precise, for a infinite polycyclic group G, given by a polycyclic presentation, we show how to determine a subgroup H of finite index in G, and construct an algorithm which carries out fast multiplication in H.