An effective version of the Lazard correspondence. (Q435965)

The Lazard correspondence provides an isomorphism between the categories of nilpotent Lie rings and finite \(p\)-groups which have the same order \(p^n\) and the same nilpotence class \(c\), for \(c<p\). The Baker-Campbell-Hausdorff formula, and its inverse, allow one to define the group operation in terms of the Lie ring ones, and vice versa. (See the book `\(p\)-automorphisms of finite \(p\)-groups' by \textit{E.~I.~Khukhro} [Lond. Math. Soc. Lect. Note Ser. 246. Cambridge: Cambridge University Press (1998; Zbl 0897.20018)].)NEWLINENEWLINE The goal of the paper under review is to develop computational methods to perform the Lazard correspondence in practice. These methods have been implemented both in MAGMA and GAP. The programmes are currently able to deal with \(p\)-groups and Lie rings of class up to \(14\).NEWLINENEWLINE Some applications are described. One is the computation of faithful modules for groups as above; a comparison is drawn with other methods. The other is the computation of Hall polynomials, which are formulas for multiplying elements, given in the so called product representation, in groups as above. We refer to the paper, which is very clearly written, for an interesting discussion of the relative advantages of this method with respect to collection, and further details.