Polynomials describing the multiplication in finitely generated torsion free nilpotent groups

TL;DR

This paper presents an algorithm to compute polynomial formulas for group operations in finitely generated torsion-free nilpotent groups, enabling explicit descriptions for groups up to Hirsch length 7.

Contribution

It introduces a method to determine Hall polynomials for all torsion-free nilpotent groups of specified Hirsch length, extending previous theoretical results.

Findings

Algorithm successfully computes Hall polynomials for groups up to Hirsch length 7.

Provides explicit polynomial descriptions for group operations in these groups.

Enhances understanding of algebraic structure of torsion-free nilpotent groups.

Abstract

A famous result of Hall asserts that the multiplication and exponentiation in finitely generated torsion free nilpotent groups can be described by rational polynomials. We describe an algorithm to determine such polynomials for all torsion free nilpotent groups of given Hirsch length. We apply this to determine the Hall polynomials for all such groups of Hirsch length at most 7.