Exercise 4, § 8

Let $c$ be an integer $\geq 1$ and let $P$ be a set of prime numbers containing all prime numbers $\leq c$. Let $\mathbf{Z}_P = S_P^{-1}\mathbf{Z}$ (§ 4, Exercise 16). Show that the terms of degree $\leq c$ in the Hausdorff series $H(U, V)$ belong to $L_{\mathbf{Z}_P}(\{U, V\})$. Deduce

† For more details on this exercise, see M. Lazard, Bull. Soc. Math. France, 91 (1963), pp. 435–451.

that, if $g$ is a nilpotent Lie $\mathbf{Z}_p$-algebra of class $\leq c$, the law of composition $(u, v) \mapsto H(u, v)$ makes $g$ into a P-torsion-free P-divisible nilpotent group of class $\leq c$. Show that conversely every group with these properties can be obtained by this process. (Use "the inversion of the Hausdorff formula", cf. § 6, Exercise 4, and show that the exponents $\alpha(m), \beta(m)$ which appear there belong to $\mathbf{Z}_p$ when $l(m) \leq c$.)

In particular, every $p$-group of order $p^n$ and class $<p$ can be obtained by means of the Hausdorff law from a nilpotent Lie $\mathbf{Z}$-algebra of class $<p$ with $p^n$ elements. (Take P to be the set of prime numbers distinct from $p$.)