An effective version of the Lazard correspondence

Abstract

The Lazard correspondence establishes an equivalence of categories between p-groups of nilpotency class less than p and nilpotent Lie rings of the same class and order. The main tools used to achieve this are the Baker–Campbell–Hausdorff formula and its inverse formulae. Here we describe methods to compute the inverse Baker–Campbell–Hausdorff formulae. Using these we get an algorithm to compute the Lie ring structure of a p-group of class < p. Furthermore, the Baker–Campbell–Hausdorff formula yields an algorithm to construct a p-group from a nilpotent Lie ring of order p n and class less than p. At the end of the paper we discuss some applications of, and practical experiences with, the algorithms.