Polynomials describing the multiplication in finitely generated 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.