We construct both free nilpotent groups and free nilpotent Lie ℚ-algebras within associative ℚ-algebras. The Baker–Hausdorff Formula is proved to be a Lie polynomial which links the operations in the group and the Lie algebra. The construction is used to embed any torsion-free nilpotent group in its ℚ-powered “hull”. In Chapter 10, all this will be applied to establish the Mal'cev Correspondence between nilpotent ℚ-powered groups and nilpotent Lie ℚ-algebras and the Lazard Correspondence for nilpotent p-groups and Lie rings of class ≤ p – 1.

Free nilpotent groups

In § 5.3 we used a free associative ℚ-algebra A to construct a free Lie ring L as a subring of A(–) We use new “calligraphic” letters for these objects, since here we prefer to denote by A = A/Ac+1 and L = L/γc+1(L) the free nilpotent factor-algebras. In this section, we construct a free nilpotent group F within A with adjoined outer unity; A is the common ground for both L and F, which helps to establish connections between them.

We recall some definitions and basic properties. Let A be a free nilpotent associative ℚ-algebra of nilpotency class c with free (non-commuting) generators x1, x2, … (when necessary, we shall take a well-ordered set of generators of any given cardinality); “nilpotent of class c” means that every product of any c + 1 elements is 0.