Introduction

The Baker–Hausdorff Formula H(x, y) is defined by the equality exey = eH(x,y) for formal power series in non-commuting variables. This formula is an important instrument in the theory of Lie groups giving a local correspondence between a Lie group and its Lie algebra, but we shall not discuss Lie groups in this paper.

The Mal'cev Correspondence makes use of the Baker–Hausdorff Formula to provide a global correspondence (a so-called equivalence of categories) between nilpotent Lie ℚ-algebras and discreet nilpotent ℚ-powered (that is, torsion-free and divisible) groups. Although finite p-groups are neither torsion-free nor divisible, we shall show how the Mal'cev Correspondence can be applied in the theory of finite p-groups.

Under some rather restrictive conditions (like the nilpotency class to be less than p) an analogous correspondence can be established between finite p-groups and Lie rings (the Lazard Correspondence). As G. Higman remarked in his talk at the Congress of Mathematicians in Edinburgh, 1958, these conditions are “… too severe to be used…, …the sort of thing one wants in the conclusion of one's theorem, rather than in the hypothesis.” Nevertheless, we shall also give examples of applications of the Lazard Correspondence. In particular, it can be used for faster reductions to Lie rings and for constructing certain examples, which may be easier for Lie rings.

As a proving ground for applications of the Baker–Hausdorff Formula, we shall discuss results on automorphisms with few fixed points, regular and almost regular ones.