Exercise 4, § 6

Let $X$ be a finite set and let $\hat{L} = \hat{L}(X)$. We give $\hat{L}$ the Hausdorff law of composition $(a, b) \mapsto a \mathbf{H} b$, which we denote simply by $a.b$. If $t \in K$, $a \in \hat{L}$, we write $a^t = ta$. (a) Let $\mathscr{H}$ be a Hall set relative to $X$. If $m \in \mathscr{H}$, let $m_L$ denote the corresponding element of the Lie algebra $L(X)$ (§ 2, no. 11) and $m_{\mathscr{H}}$ the basic commutator of the group $\hat{L}$ defined by $m$ (§ 5, no. 4, Remark). If $m$ is of length $l$, show that $m_{\mathscr{H}} = m_L + \mu$, where $\mu$ is of filtration $\geqslant l + 1$ (argue by induction on $l$). Deduce that every element $w$ of the group $\hat{L}$ can be written uniquely in the form

where $\alpha(m) \in K$ and the product is convergent in the topological group $\hat{L}$.

(b) Let $P$ be the set of prime numbers and let $c$ be an integer $\geqslant 1$. Let $K = \mathbf{Q}$. Show that the group $\hat{L}/\hat{L}_c$ is identified with the $P$-envelope $F_P^c$ of the group $F^c = F(X)/C^cF(X)$, cf. § 5, Exercise 6.

(c) Take $X$ to be a set of two elements $\{U, V\}$. Show the existence of two families $\alpha(m), \beta(m)$ of rational numbers such that

For example

(d) Let $g$ be a nilpotent Lie $\mathbf{Q}$-algebra with the Hausdorff group law. Let $u, v$ be in $g$. Show that

where $\alpha$ and $\beta$ are the families of rational numbers defined above ("inversion of the Hausdorff formula"). (Use the continuous homomorphism $\phi : \hat{L} \to g$ such that $\phi(U) = u, \phi(V) = v$ and note that it is also a homomorphism for the Hausdorff group law.)

(e) Let $G$ be a torsion-free divisible nilpotent group (i.e. $P$-torsion-free and $P$-divisible). If $u \in G, t \in \mathbf{Q}, u^t$ is defined (§ 4, Exercise 16). If $u \in G, v \in G$, we define $u + v$ and $[u, v]$ by the formulae of (d) above. Show that $G$ thus has a nilpotent Lie $\mathbf{Q}$-algebra structure and that the corresponding Hausdorff law is the given group law on $G$ (verify these assertions when $G$ is a group $F_P^c$, cf. (b), and pass from this to the general case using homomorphisms of $F_P^c$ into $G$).

Let $f$ be a mapping of $G$ into a torsion-free divisible nilpotent group $G'$. Show that $f$ is a (group) homomorphism if and only if it is a Lie algebra homomorphism. (The Hausdorff law therefore defines an isomorphism of the "category" of nilpotent Lie $\mathbf{Q}$-algebras onto that of torsion-free divisible nilpotent groups.)